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Research ArticleFractional Quantum Field Theory From Lattice to Continuum

Vasily E Tarasov

Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University Moscow 119991 Russia

Correspondence should be addressed to Vasily E Tarasov tarasovtheorysinpmsuru

Received 23 August 2014 Accepted 8 October 2014 Published 30 October 2014

Academic Editor Elias C Vagenas

Copyright copy 2014 Vasily E Tarasov This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

An approach to formulate fractional field theories on unbounded lattice space-time is suggested A fractional-order analog ofthe lattice quantum field theories is considered Lattice analogs of the fractional-order 4-dimensional differential operators areproposed We prove that continuum limit of the suggested lattice field theory gives a fractional field theory for the continuum4-dimensional space-time The fractional field equations which are derived from equations for lattice space-time with long-rangeproperties of power-law type contain the Riesz type derivatives on noninteger orders with respect to space-time coordinates

1 Introduction

Fractional calculus and fractional differential equations [12] have a wide application in mechanics and physics Thetheory of integrodifferential equations of noninteger ordersis powerful tool to describe the dynamics of systems andprocesses with power-law nonlocality long-range memoryandor fractal properties Recently the spatial fractional-order derivatives have been actively used in the space-fractional quantummechanics suggested in [3 4] the quanti-zation of fractional derivatives [5] the fractional Heisenbergand quantum Markovian equations [6 7] the fractional the-ory of open quantum systems [8 9] the quantum field theoryand gravity for fractional space-time [10 11] and the frac-tional quantum field theory at positive temperature [12 13]Fractional calculus allows us to take into account fractionalpower-law nonlocality of continuously distributed systemsUsing the fractional calculus we can consider space-timefractional differential equations in the quantum field theoryThe fractional-order Laplace and drsquoAlembert operators havebeen suggested by Riesz in [14] in 1949 for the first timeThen noninteger powers of drsquoAlembertian are considered indifferent works (for example see Section 28 in [1] and [15ndash17]) The fractional Laplace and drsquoAlembert operators of theby Riesz type are a base in the construction of the fractionalfield theory in multidimensional spaces As it was shown in[18ndash20] the continuum equations with fractional derivatives

of the Riesz type can be directly connected to lattice mod-els with long-range properties A connection between thedynamics of lattice system with long-range properties andthe fractional continuum equations is proved by using thetransform operation [18ndash20] and it has been applied for themedia with spatial dispersion law [21 22] for the fields infractional nonlocal materials [23 24] for fractional statisticalmechanics [25] and for nonlinear classical fields [26]

A characteristic feature of continuum quantum fieldtheories which are used in high-energy physics is ultravioletdivergences The divergences arise in momentum (wave-vector) space from modes of very high wave number thatis the structure of the field theories at very short distancesIn the narrow class of quantum theories which are calledldquorenormalizablerdquo the divergences can be removed by a singu-lar redefinition of the parameters of the theory This processis called the renormalization [27] and it defines a quantumfield theory as a nontrivial limit of theory with an ultravioletcut-off The renormalization requires the regularization ofthe path integrals in momentum space These regularizedintegrals depend on parameters such as the momentum cut-off the Pauli-Villars masses and the dimensional regulariza-tion parameter which are used in the correspondent regular-ization procedure This regularized integration is ultravioletfinite In some sense we can say that the regularization pro-cedure consists in the introduction of a momentum cut-off

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 957863 14 pageshttpdxdoiorg1011552014957863

2 Advances in High Energy Physics

In quantum field theory the path integral approach isvery important to describe processes in high energy physics[28] The path integrals are well-defined for systems with adenumerable number of degrees of freedom In field theorywe are dealing with the case of an innumerable number ofdegrees of freedom labeled by the space-time coordinates atleast To give the path integrals a precisemeaning we can dis-cretize space and time that is we can introduce a space-timelattice The introduction of a lattice space-time correspondsto a special form of regularization of the path integrals Inthe lattice field theory the momentum space integrals will becut off at a momentum of the order of the inverse lattice con-stants The lattice regularization can be considered as a natu-ral introduction of amomentum cut-offThe lattice renorma-lization procedure can be carried out for path integrals inmomentum space The first step of the procedure is regu-larization that consists in introducing a space-time latticeThis regularization allows us to give an exact definition ofthe path integral since the lattice has the denumerable num-ber of degrees of freedom Moreover the existence of themomentum cut-off is not surprising in the lattice fieldtheories In the expression of the Fourier integral for latticefields themomentum integrationwith respect towave-vectorcomponents 119896120583 (120583 = 1 2 3 4) is restricted by the Brillouinzone 119896120583 isin [minus120587119886120583 120587119886120583] where 119886120583 are the lattice constantsThe second step of the renormalization is a continualizationprocedure that removes the lattice structure by a continuumlimit where the lattice constants 119886120583 tend to zero In thisstep of the renormalization process the momentum cut-offis removed by the continuum limit

At the present time fractional-order generalization of thelattice field theories has not been suggested Lattice approachto the fractional field theories was not previously consideredIn this paper we propose a formulation of fractional fieldtheory on a lattice space-time The suggested theory can beconsidered as a lattice analog of the fractional field theoriesand as a fractional-order analog of the lattice quantum fieldtheories For simplification we consider the free scalar fieldsIt allows us to demonstrate a number of important propertiesin details The lattice analogs of the fractional-order differen-tial operators are suggestedWe prove that continuum limit ofthe suggested lattice theory gives the fractional field theorywith continuum space-time The fractional field equationscontain the Riesz type derivatives on noninteger orders withrespect to space-time coordinates In Section 2 the fractionalfield theory on continuum space-time is considered for scalarfields and fractional-order differential operators are defined

In Section 3 the fractional-order lattice differential oper-ators of noninteger orders are considered and the latticefractional field theory for lattice space-time is proposed InSection 4 the lattice-continuum transformation of the latticefractional theories is discussed A short conclusion is given inSection 5

2 Fractional Field Theory onContinuum Space-Time

21 Scalar Field in Pseudo-Euclidean Space-Time For sim-plification we consider scalar fields in the 4-dimensional

pseudo-Euclidean space-timeR413 Let us consider the classi-

cal field equation

(◻ +1198722) 120593 (119909) = 119869 (119909) (1)

where ◻ is the drsquoAlembert operator 120593(119909) is a real field and119909 isin R4

13is the space-time vector with components 119909120583 where

120583 = 0 1 2 3 This field equation follows from stationaryaction principle 120575119878[120593] = 0 where the action 119878[120593] has theform

119878 [120593] = minus1

2int1198894119909120593 (119909) (◻ +119872

2) 120593 (119909) (2)

In the quantum theory the generalized coordinates 120593(119909)and momenta (119909) become the operators Φ(119909) and Φ(119909)that satisfy the canonical commutation relations The Greenfunctions are

119866119904 (1199091 119909119904) = ⟨Ω1003816100381610038161003816119879 Φ (1199091) sdot sdot sdot Φ (119909119904)

1003816100381610038161003816 Ω⟩ (3)

where |Ω⟩ denotes the ground state ldquophysical vacuumrdquo ofthe fields and 119879 denotes the time-ordered product of theoperators Φ(119909) These Green functions have a path integralrepresentation [28] in the form

119866119904 (1199091 119909119904) =int119863120593 (120593 (1199091) sdot sdot sdot 120593 (119909119904)) 119890

119894119878[120593]

int119863120593119890119894119878[120593] (4)

where int119863120593 is the sum over all possible configurations ofthe field 120593(119909) The effects arising from quantum fluctuationsare defined by those contributions to the integral (4) thatcome from field configurations which are not solutions ofthe classical field equation (1) and hence do not lead to astationary action

22 From Pseudo-Euclidean to Euclidean Space-Time Let usconsider the analytic continuation of (4) to imaginary times(see Section 74 in [29]) such that

1199090997888rarr minus1198941199094 (5)

We will use 119909119864 to denote the Euclidean four-vector Itallows us to consider the Euclidean space-time R4 instead ofthe pseudo-Euclidean space-time R4

13

The Euclidean action 119878119864[120593119888] is obtained from the action(2) by using the three steps

(1) firstly the replacement 1199090 rarr minus1198941199094 where 1199090 appears

explicitly(2) the use of the real valued field 120593119888(119909119864) = 120593119888( 1199094)

instead of120593( 119905) where120593119888( 1199094) is not obtained from120593(119905 ) by substituting 1199094 for 119905 noting that 120593119888(119909119864) =

120593119888( 1199094) is a real field which is a function of theEuclidean variable 119909119864

(3) multiplying the resultant expression by minus119894

As a result this leads to the expression

119878119864 [120593119888] =1

2int1198894119909119864120593119888 (119909119864) (minus◻119864 +119872

2) 120593119888 (119909119864) (6)

Advances in High Energy Physics 3

where ◻119864 denotes the 4-dimensional Laplacian

◻119864 =

4

sum

120583=1

1205972

1205971199092119864120583

(7)

To formulate a fractional generalization of the field the-ory we should use the physically dimensionless space-timecoordinates It allows us to have the samephysical dimensionsfor all other physical values as in the usual (nonfractional)field theories with dimensionless coordinates In this paperall the quantities will be done physically dimensionless tosimplify our consideration For this aim we scale the massparameter 119872 the coordinates 119909119864 and the field 120593119888 accordingto their physical dimension As seen from (6) the quantities120593119888 and 119872 have the physical dimension that is the inverselength and it is obviously 119909119864 that has the dimension of lengthsince the action is dimensionlessTherefore we can define thedimensionless quantities119872119862 x and 120593119862 by the replacement

x = 119897minus1

0119909119864 119909120583 = 119897

minus1

0119909119864120583 120593119862 = 119897

minus1

0120593119888

119872119862 = 119897minus1

0119872

(8)

We will use x (instead of 119909119864) to denote the Euclidean four-vector with components 119909120583 where 120583 = 1 2 3 4 As a resultthis leads to the expression for the Euclidean action 119878119864[120593119862] isgiven by the equation

119878119864 [120593119862] =1

2int1198894x120593119862 (x) (minus◻119864119862 +119872

2

119862) 120593119862 (x) (9)

where ◻119864119862 denotes the 4-dimensional Laplacian for dimen-sionless variables x of continuum space-time such that

◻119864119862 =

4

sum

120583=1

1205972

1205971199092120583

(10)

The Green functions (3) which are continued to imagi-nary times have the path integral representation

⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩119864 =int119863120593119862 (120593119862 (x1) sdot sdot sdot 120593119862 (x119904)) 119890minus119878119864[120593119862]

int119863120593119890minus119878119864[120593119862]

(11)

where we use the notation for the Euclidean Green functionfor physically dimensionless quantities (8)

In the imaginary time formulation of quantum field the-ory the Green functions look like the correlation functionsused in statistical mechanics The partition function has theform

119885 = int119863120593119862 (x) 119890minus119878119864[120593119862] (12)

where the integration measure119863120593119862 is formally defined by

119863120593119862 = prodx119889120593119862 (x) (13)

Most of the variables of the system can be expressed in termsof the partition function or its derivatives

23 Continuum Fractional Derivatives of the Riesz Type Toformulate a fractional generalization of the quantum fieldtheory we define fractional-order derivatives with respect todimensionless Euclidean coordinates 119909120583 where 120583 = 1 2 3 4These derivatives will be denoted by Dplusmn

119862[120572120583 ] where 120572 is the

order of the derivative 120583 denotes the coordinate 119909120583 withrespect to which the derivative is taken 119862 marks that thederivative is used for continuum field theory (119871 will be usedfor lattice operators) and + and minus denote the even and oddtypes of the derivatives

Definition 1 Continuum fractional derivatives D+119862[120572120583 ] of the

Riesz type and noninteger order 120572 gt 0 are defined by theequation

D+

119862[120572

120583]120593119862 (x) =

1

1198891 (119898 120572)intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572+1(Δ119898

119911120583120593) (x) 119889119911120583

(0 lt 120572 lt 119898)

(14)

where 119898 is the integer number that is greater than 120572 and theoperators (Δ119898

119911120583120593)(x) are a finite difference [1 2] of order119898 of

a function 120593119862(x) with the vector step z120583 = 119911120583e120583 isin R4 for thepoint x isin R4 The centered difference

(Δ119898

119911120583120593) (x120583) =

119898

sum

119899=0

(minus1)119899 119898

119899 (119898 minus 119899)120593 (x minus (

119898

2minus 119899) 119911120583e120583)

(15)

The constant 1198891(119898 120572) is defined by

1198891 (119898 120572) =12058732

119860119898 (120572)

2120572Γ (1 + 1205722) Γ ((1 + 120572) 2) sin (1205871205722) (16)

where

119860119898 (120572) = 2

[1198982]

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)(119898

2minus 119895)

120572

(17)

for the centered difference (15)

The constants 1198891(119898 120572) are different from zero for all 120572 gt

0 in the case of an even 119898 and centered difference (Δ119898

119894119906)

(see Theorem 261 in [1]) Note that the integral (14) does notdepend on the choice of 119898 gt 120572 Therefore we can alwayschoose an even number119898 so that it is greater than parameter120572 and we can use the centered difference (15) for all positivereal values of 120572

Using (14) we can see that the continuum fractionalderivative D+

119862[120572120583 ] is the Riesz derivative that acts on the field

120593119862(x) with respect to the component 119909120583 isin R1 of the vectorx isin R4 that is the operator D+

119862[120572120583 ] can be considered as a

partial fractional derivative of Riesz typeThe Riesz fractional derivatives for even 120572 = 2119898 where

119898 isin N are connected with the usual partial derivative ofinteger orders 2119898 by the relation

D+119862[2119898

120583] 120593119862 (x) = (minus1)

119898 1205972119898

120593119862 (x)1205971199092119898

120583

(18)


The fractional derivativesD+119862[2119898120583 ] for even orders 120572 are local

operators Note that the Riesz derivative D+119862[1120583 ] cannot be

considered as a derivative of first order with respect to 119909120583that is

D+

119862[1

120583] 120593119862 (x) =

120597120593119862 (x)120597119909120583

(19)

For 120572 = 1 the operator D+119862[1120583 ] is nonlocal like a ldquosquare

root of the Laplacianrdquo Note that the Riesz derivatives for oddorders 120572 = 2119898 + 1 where119898 isin N are nonlocal operators thatcannot be considered as usual derivatives 1205972119898+11205971199092119898+1

An important property of the Riesz fractional derivativesis the Fourier transformF of these operators in the form

F(D+

119862[120572

120583]120593119862 (x)) (k) = 10038161003816100381610038161003816

11989612058310038161003816100381610038161003816

120572

(F120593) (k) (20)

Property (20) is valid for functions 120593119862(x) from the space ofinfinitely differentiable functions with compact support Italso holds for the Lizorkin space (see Section 81 in [1])

Let us consider the continuum fractional derivativeDminus119862[120572120583 ] of the Riesz type that has the property

F(Dminus

119862[120572

120583]120593119862 (x)) (k) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

(F120593) (k)

(120572 gt 0)

(21)

where sgn(119896120583) is the sign function that extracts the sign of areal number (119896120583) For 0 lt 120572 lt 1 the operator Dminus

119862[120572120583 ] can be

considered as the conjugate Riesz derivative [30] with respectto 119909120583 Therefore the operator (21) will be called a generalizedconjugate derivative of the Riesz type

The fractional operator Dminus119862[120572120583 ] will be defined separately

for the following three cases (a) 120572 gt 1 (b) 120572 = 1 (c) 0 lt 120572 lt

1

Definition 2 Continuum fractional derivativesD+119862[120572120583 ] of the

Riesz type are defined by the following equations

(a) For 120572 gt 1 the fractional operator D+119862[120572120583 ] is defined

by the equation

Dminus119862

[120572

119895] 120593119862 (x)

=1

1198891 (119898 120572 minus 1)

120597

120597119909120583

intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572 (Δ119898

119911120583120593) (x) 119889119911120583

(1 lt 120572 lt 119898 + 1)

(22)

where (Δ119898119911120583120593)(x) is a finite difference that is defined in (15)

(b) For integer values 120572 = 1 we have

Dminus119862

[1

120583]120593119862 (x) =

120597120593119862 (x)120597119909120583

(23)

(c) For 0 lt 120572 lt 1 the fractional operator D+119862[120572120583 ] is

defined by the equation

Dminus

119862[120572

120583]120593119862 (x)

=120597

120597119909120583

intR1

1198771minus120572 (119909120583 minus 119911120583) 120593 (x + (119911120583 minus 119909120583) e120583) 119889119911120583

(0 lt 120572 lt 1)

(24)

where e120583 is the basis of the Cartesian coordinate system thefunction 119877120572(119909) is the Riesz kernel that is defined by

119877120572 (119909) =

120574minus1

1(120572) |119909|

120572minus1120572 = 2119899 + 1 119899 isin N

minus120574minus1

1(120572) |119909|

120572minus1 ln |119909| 120572 = 2119899 + 1 119899 isin N

(25)

and the constant 1205741(120572) has the form

1205741 (120572)

=

212057212058712

Γ (1205722)

Γ ((1 minus 120572) 2)120572 = 2119899 + 1

(minus1)(1minus120572)2

2120572minus1

12058712

Γ (120572

2) Γ (1 +

[120572 minus 1]

2) 120572 = 2119899 + 1

(26)

with 119899 isin N and 120572 isin R+

Note that the distinction between the continuum frac-tional derivatives Dminus

119862[120572120583 ] and the Riesz 4-dimensional frac-

tional derivative consists [2] in the use of |119896120583|120572 instead of |k|120572

For integer odd values of 120572 we have

Dminus

119862[2119898 + 1

120583]120593119862 (x) = (minus1)

119898 1205972119898+1

120593119862 (x)1205971199092119898+1

120583

(119898 isin N)

(27)

Equation (27) means that the fractional derivativesDminus119862[120572120583 ] of the odd orders 120572 are local operators represented

by the usual derivatives of integer orders Note that thecontinuum derivative Dminus

119862[2119898120583 ] with 119898 isin N cannot be

considered as a local derivative of the order 2119898 with respectto 119909120583 For 120572 = 2 the generalized conjugate Riesz derivative isnot the local derivative 120597

21205972119909120583 The derivatives Dminus

119862[120572120583 ] for

even orders 120572 = 2119898 where 119898 isin N are nonlocal operatorsthat cannot be considered as usual derivatives 12059721198981205971199092119898

120583

It is important to note that the usual Leibniz rule for thederivative of products of two ormore functions does not holdfor derivatives of noninteger orders and for integer ordersdifferent from one [31] This violation of the usual Leibnizrule is a characteristic property of all types of fractionalderivatives

Equations (18) and (27) allow us to state that the partialderivatives of integer orders are obtained from the fractional


derivatives of the Riesz typeDplusmn119862[120572120583 ] for odd values 120572 = 2119898119895+

1 gt 0 by Dminus119862[120572120583 ] only and for even values 120572 = 2119898 gt 0 (119898 isin

N) by D+119862[120572120583 ] The continuum derivatives of the Riesz type

Dminus119862[2119898120583 ] andD+119862 [

2119898+1120583 ] are nonlocal differential operators of

integer ordersIn formulation of fractional analogs of classical field theo-

ries we need to generalize some field equations with partialdifferential equations of integer order It is obvious that wewould like to have a fractional generalization of these integer-order differential equations so as to obtain the originalequations in the limit case when the orders of generalizedderivatives become equal to initial integer values In orderfor this requirement to hold we can use the following rulesof generalization

1205972119898

1205971199092119898120583

= (minus1)119898D+119862 [

2119898

120583] 997888rarr (minus1)

119898D+

119862[120572

120583]

(119898 isin N 2119898 minus 1 lt 120572 lt 2119898 + 1)

1205972119898+1

1205971199092119898+1120583

= (minus1)119898Dminus119862[2119895 + 1

120583] 997888rarr (minus1)

119898Dminus

119862[120572

120583]

(119898 isin N 2119898 lt 120572 lt 2119898 + 2)

(28)

In order to derive a fractional generalization of differentialequation with partial derivatives of integer orders we shouldreplace the usual derivatives of odd orders with respect to 119909120583

by the continuum fractional derivativesDminus119862[120572120583 ] and the usual

derivatives of even orderswith respect to119909120583 by the continuumfractional derivatives of the Riesz type D+

119862[120572120583 ]

24 Continuum Fractional 4-Dimensional Laplacian The 4-dimensional Laplacian ◻119864119862 is defined by (10) as an operatorof second order for Euclidean space-time

Fractional-order generalizations of the drsquoAlembert oper-ator ◻ and the119873-dimensional Laplacian ◻119864 are considered in[14] and in Section 28 of [1]

It is important to note that an action of two repeatedfractional derivatives of order 120572 is not equivalent to the actionof the fractional derivative of the double order 2120572

Dplusmn

119862[120572

120583]Dplusmn

119862[120572

120583] = D

plusmn

119862[2120572

120583] (120572 gt 0) (29)

The continuum 4-dimensional Laplacian of nonintegerorder for the scalar field 120593119862(x) can be defined by two differentequations where the first expression contains the two latticeoperators of order 120572 and the second expression contains thefractional derivatives of the doubled order 2120572

Definition 3 The continuum 4-dimensional Laplace opera-tors ◻120572120572plusmn

119864119862and ◻

2120572plusmn

119864119862of noninteger order 2120572 for the scalar field

120593119862(x) are defined by the different equations

◻120572120572plusmn

119864119862120593119862 (x) =

4

sum

120583=1

(Dplusmn

119862[120572

120583])

2

120593119862 (x) (30)

◻2120572plusmn

119864119862120593119871 (x) =

4

sum

120583=1

Dplusmn

119862[2120572

120583]120593119862 (x) (31)

where Dplusmn119862are defined in Definitions 1 and 2

The violation of the semigroup property (29) leads to thefact that the operators (30) and (31) donot coincide in general

It should be noted that the operators ◻120572120572minus119864119862

and ◻2120572+

119864119862for

integer 120572 = 1 gives the usual (local) 4-dimensional Laplacian◻119864 that is defined by (7) that is

◻11minus

119864119862= ◻2+

119864119862= ◻119864 (32)

The operators ◻120572120572+119864119862

and ◻2120572minus

119864119862for integer 120572 = 1 are non-

local operators of the second orders that cannot be consideredas ◻119864

◻11+

119864119862= ◻119864 ◻

2minus

119864119862= ◻119864 (33)

Therefore we should use only the continuum fractional 4-dimensional Laplace operators◻120572120572minus

119864119862or◻2120572+119864119862

in the fractionalfield theory since the operators ◻120572120572+

119864119862or ◻2120572minus119864119862

do not satisfythe correspondence principle for 120572 = 1

Fractional Laplace operators have been suggested byRiesz in [14] for the first time The fractional Laplacian(minusΔ)1205722

119862in the Riesz form for 4-dimensional Euclidean space-

time R4 can be considered as an inverse Fourierrsquos integraltransformFminus1 of |k|120572 by

((minusΔ)1205722

119862120593) (x) = F

minus1(|k|120572 (F120593) (k)) (34)

where 120572 gt 0 and x isin R4

Definition 4 For 120572 gt 0 the fractional Laplacian of the Rieszform is defined as the hypersingular integral

((minusΔ)1205722

119862120593119862) (x) =

1

1198894 (119898 120572)intR4

1

|z|120572+4(Δ119898

z120593119862) (z) 1198894z

(35)

where 119898 gt 120572 and (Δ119898

z120593)(z) is a finite difference of order 119898of a field 120593119862(x) with a vector step z isin R4 and centered at thepoint x isin R4

(Δ119898

z120593) (z) =119898

sum

119895=0

(minus1)119895 119898

119895 (119898 minus 119895)120593 (x minus 119895z) (36)


1198894 (119898 120572) =1205873119860119898 (120572)

2120572Γ (1 + 1205722) Γ (2 + 1205722) sin (1205871205722) (37)

where

119860119898 (120572) =

119898

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)119895120572 (38)

Note that the hypersingular integral (35) does not dependon the choice of 119898 gt 120572 The Fourier transform F ofthe fractional Laplacian is given by F(minusΔ)

1205722

119862120593(k) =

|k|120572(F120593)(k) This equation is valid for the Lizorkin space [1]


and the space119862infin(R4) of infinitely differentiable functions onR4 with compact support

25 Fractional Field Equations The Euclidean action 119878119864[120593119862]

for fractional scalar fields can be defined by the expression

119878(120572)

119864[120593119862 119869119862]

=1

2int1198894x120593119862 (x) (◻

2120572+

119864119862+1198722

119862) 120593119862 (x) + int119889

4x119869119862 (x) 120593119862 (x) (39)

where ◻2120572+

119864119862denotes the fractional 4-dimensional Laplacian

(31) for dimensionless variables x of continuum space-timeHere we take into account (18) in the form ◻

2+

119864119862= minus◻119864119862

Using the stationary action principle 120575119878(120572)119864

[120593119862 119869119862] = 0we derive the fractional field equation

(◻2120572+

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x) (40)

Similarly we can consider the fractional field theories that aredescribed by the fractional field equations

(◻120572120572minus

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

((minusΔ)1205722

119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

(41)

where the fractional 4-dimensional Laplacians (30) and (35)are used

The Green functions 119866(120572)

119904119862119864(x1 x119904) = ⟨120593119862(x1) sdot sdot sdot

120593119862(x119904)⟩(120572)

119864for Euclidean space-time and dimensionless vari-

ables have the following path integral representation

119866(120572)

119904119862119864(x1 x119904) =

int119863120593119862 (120593119862 (x1) sdot sdot sdot 120593119862 (x119904)) 119890minus119878(120572)

119864[120593119862119869119862]

int119863120593119862119890minus119878(120572)

119864[120593119862119869119862]

(42)

where int119863120593119862 is the sum over all possible configurations ofthe field 120593119862(x) for continuum space-time Note that the path-integral approach for space-fractional quantummechanics isconsidered in [3 4 32]

The Euclidean Green functions (42) of fractional fieldtheory can be derived from the generating functional

119885(120572)

0119862[119869119862] = int119863120593119862119890

minus119878(120572)

119864[120593119862119869119862] (43)

Using the integer-order differentiation of (43) with respect tothe sources 119869119899 we can obtain the correlation functions The119904-point fractional correlation function is

⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩(120572)

119864=

120575119904119885(120572)

0119862[119869119862]

120575119869119862 (x1) sdot sdot sdot 120575119869119862 (x119904) (44)

Quantum fluctuations correspond to the contributions tothe integral (43) coming from field configurations which arenot solutions to the classical field equations (40) and (41)

3 Fractional Field Theory onLattice Space-Time

31 Lattice Space-Time In quantum field theory a latticeapproach is based on lattice space-time instead of thecontinuum of space-time Lattice models originally occurredin the condensed matter physics where the atoms of a crystalform a lattice The unit cell is represented in terms of thelattice parameters which are the lengths of the cell edges (a120583where 120583 = 1 2 3 4) and the angles between them

Let us consider an unbounded space-time lattice charac-terized by the noncoplanar vectors a120583 120583 = 1 2 3 4 that arethe shortest vectors by which a lattice can be displaced andbe brought back into itself For simplification we assume thata120583 120583 = 1 2 3 4 are mutually perpendicular primitive latticevectors We choose directions of the axes of the Cartesiancoordinate system coinciding with the vector a120583 Then a120583 =119886120583e120583 where 119886120583 = |a120583| and e120583 (120583 = 1 2 3 4) are thebasis vectors of theCartesian coordinate system for Euclideanspace-time R4 This simplification means that the latticeis a primitive 4-dimensional orthorhombic Bravais latticeThe position vector of an arbitrary lattice site is writtenas

x (n) =4

sum

120583=1

119899120583a120583 (45)

where 119899120583 are integer In a lattice the sites are numbered by nso that the vector n = (1198991 1198992 1198994 1198994) can be considered as anumber vector of the corresponding lattice site

As the lattice fields we consider real-valued functions forn-sites For simplification we consider the scalar field 120593119871(n)for lattice sites that is defined by n = (1198991 1198992 1198993 1198994) In manycases we can assume that120593119871(n) belongs to theHilbert space 1198972of square-summable sequences to apply the discrete Fouriertransform For simplification we will consider operatorsfor the lattice scalar fields 120593119871(n) = 120593(1198991 1198992 1198993 1198994) Allconsideration can be easily generalized to the case of thevector fields and other types of fields

For continuum fractional field theory we use the dimen-sionless quantities (8) In the lattice fractional theory we alsowill be using the physically dimensionless quantities such as119886120583 119899120583 x(n) e120583 and 120593119871(n)

32 Lattice Fractional Derivative Let us give a definitionof lattice partial derivative Dplusmn

119871[120572120583 ] of arbitrary positive real

order 120572 in the direction e120583 = a120583|a120583| in the lattice space-time

Definition 5 Lattice fractional partial derivatives are theoperators Dplusmn

119871[120572120583 ] such that

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum

119898120583=minusinfin

119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(120583 = 1 2 3 4)

(46)


where 120572 isin R 120572 gt 0 119899120583 119898120583 isin Z and the kernels 119870plusmn120572(119899 minus 119898)

are defined by the equations

119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)

where11198652 is the Gauss hypergeometric function [33 34]

The parameter 120572 gt 0 will be called the order of the latticederivatives (46)

The kernels 119870plusmn

120572(119899) are real-valued functions of integer

variable 119899 isin Z The kernel 119870+120572(119899) is even function 119870

+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all

119899 isin ZThe reasons to define the kernels 119870plusmn

120572(119899 minus 119898) in the forms

(47) and (48) are based on the expressions of their Fourierseries transforms The Fourier series transform

+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)

for the kernel119870+120572(119899) defined by (47) satisfies the condition

+

120572(119896) = |119896|

120572 (120572 gt 0) (50)

The Fourier series transforms

minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)

for the kernels119870minus120572(119899) defined by (48) satisfies the condition

minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)

Note that we use the minus sign in the exponents of (49) and(51) instead of plus in order to have the plus sign for planewaves and for the Fourier series

The form (47) of the kernel 119870+120572(119899 minus 119898) is completely

determined by the requirement (50) If we use an inverserelation of (49) with

+

120572(119896) = |119896|

120572 that has the form

119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)

then we get (47) for the kernel 119870+120572(119899 minus 119898) The form (48) of

the term 119870minus

120572(119899 minus 119898) is completely determined by (52) Using

the inverse relation of (51) with minus

120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)

we get (48) for the kernel 119870minus120572(119899 minus 119898) Note that119870minus

120572(0) = 0

The lattice operators (46) with (47) and (48) for integerand noninteger orders 120572 can be interpreted as a long-rangeinteractions of the lattice site defined by 119899 with all other siteswith119898 = 119899

33 Lattice Operators of Integer Orders Let us give exactforms of the kernels plusmn

120572(119896) for integer positive 120572 isin N Equa-

tions (47) and (48) for the case 120572 isin N can be simplifiedTo obtain the simplified expressions for kernels plusmn

120572(119896) with

positive integer 120572 = 119898 we use the integrals of Sec 2535 in[35]The kernels119870plusmn

120572(119899) for integer positive 120572 = 119898 are defined

by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2

minus(minus1)[1205722]

119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)

where [119909] is the integer part of the value 119909 and 119899 isin N Here2[(119898 + 1)2] minus 119898 = 1 for odd 119898 and 2[(119898 + 1)2] minus 119898 = 0

for even119898Using (55) or direct integration (53) for integer values 120572 =

1 and120572 = 2 we get the simplest examples of119870+120572(119899) in the form

119870+

1(119899) = minus

1 minus (minus1)119899

1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N

Using (56) or direct integration (54) for 120572 = 1 and 120572 = 2 weget examples of119870minus

120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)

where 119899 = 0 119899 isin Z and 119870minus

119898(0) = 0 for all 119898 isin N Note that

(1 minus (minus1)119899) = 2 for odd 119899 and (1 minus (minus1)

119899) = 0 for even 119899

In the definition of lattice fractional derivatives (46) thevalue 120583 = 1 2 3 4 characterizes the component 119899120583 of thelattice vector n with respect to which this derivative is takenIt is similar to the variable 119909120583 in the usual partial derivativesfor the space-time R4 The lattice operators Dplusmn

119871[120572120583 ] are

analogous to the partial derivatives of order 120572 with respectto coordinates 119909120583 for continuum field theory The latticederivativeDplusmn

119871[120572120583 ] is an operator along the vector e120583 = a120583|a120583|

in the lattice space-time


34 Lattice Operators with Other Kernels In general we canweaken the conditions (50) and (52) to determine a morewider class of the lattice fractional derivatives For this aimwe replace the exact conditions (50) and (52) by the asympto-tical requirements

+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)

where the little-o notation 119900(|119896|120572) means the terms that

include higher powers of |119896| than |119896|120572 The conditions (59)

and (60) mean that we can consider arbitrary functions119870plusmn

120572(119899 minus 119898) for which

plusmn

120572(119896) are asymptotically equivalent to

|119896|120572 and 119894 sgn(119896)|119896|120572 as |119896| rarr 0 respectivelyAs an example of the kernel 119870+

120572(119899 minus 119898) which can give

the lattice fractional derivatives (46) with (59) has been sug-gested in [18ndash20] in the form

119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)

where we use relation 54812 from [35]This kernel has beensuggested in [18 19] to describe long-range interactions of thelattice particles for noninteger values of 120572 For integer valuesof 120572 isin N the kernel 119870+

120572(119899 minus 119898) = 0 for |119899 minus 119898| ge 1205722 +

1 For 120572 = 2119895 we have 119870+

120572(119899 minus 119898) = 0 for all |119899 minus 119898| ge

119895 + 1 The function 119870+

120572(119899 minus 119898) with even value of 120572 = 2119895

can be interpreted as an interaction of the 119899-particle with 2119895

particles with numbers 119899plusmn1 sdot sdot sdot 119899plusmn119895 Note that the long-rangeinteractionwith the kernel (61) is partially connectedwith thelong-range interaction of the Grunwald-Letnikov-Riesz type[24] It is easy to see that expression (47) is more complicatedthan (61)

As an example of the kernel 119870minus120572(119899 minus 119898) which can give

the lattice fractional derivatives (46) with (60) has beensuggested in [20] in the form

119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)

where the brackets [ ] mean the integral part that is thefloor function that maps a real number to the largest previousinteger number The expression (2[(119899 + 1)2] minus 119899) is equal tozero for even 119899 = 2119898 and it is equal to 1 for odd 119899 = 2119898 minus 1To get the expression we use relation 54813 from [35] Notethat the kernel (62) is real valued function since we have zerowhen the expression (minus1)

(119899+1)2 becomes a complex numberFor 0 lt 120572 le 2 we can give other examples of the kernels

with the property (59) which are given in Section 8 of thebook [36] For example the most frequently used kernel is

119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)

where we use the multiplier 119860(120572) = (2Γ(minus120572) cos(1205871205722))minus1which has the asymptotic behavior +

120572(119896) =

+

120572(0) + |119896|

120572+

119900(|119896|120572) (119896 rarr 0) for the cases 0 lt 120572 lt 2 and 120572 = 1

with nonzero term +

120572(0) where 120577(119911) is the Riemann zeta-

function To take into account this expression we use theasymptotic condition for +

120572(119896) in the form (50) that includes

+

120572(0) For details see Section 811-812 in [36]

35 Lattice Fractional 4-Dimensional Laplacian An action oftwo repeated lattice operators of order 120572 is not equivalent tothe action of the lattice operator of double order 2120572

Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)

Note that these properties are similar to noninteger orderderivatives [2]

Definition 6 The lattice 4-dimensional fractional Laplacianoperators ◻

120572120572plusmn

119864119871and ◻

2120572plusmn

119864119871for a scalar lattice field 120593119871(m)

are defined by the following two equations where the firstexpression contains the two lattice operators of order 120572

◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)

and the second expression contains the lattice operator of theorder 2120572 in the form

◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)

The violation of the semigroup property (64) leads to thefact that operators (65) and (66) do not coincide in general

Using (46) expression (66) can be represented by

(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)

The correspondent continuum fractional Laplace opera-tors are defined by (30) and (31) The continuum operators◻120572120572minus

119864119862and ◻

2120572+

119864119862for integer 120572 = 1 give the usual (local) 4-

dimensional Laplacian◻119864 that is defined by (7)Theoperators◻120572120572+

119864119862and ◻

2120572minus

119864119862for integer 120572 = 1 are nonlocal operators and

cannot get a correspondence with the usual (nonfractional)field theories Therefore we should use the lattice fractionalLaplace operators ◻120572120572minus

119864119871or ◻2120572+119864119871

in the lattice fractional fieldtheories

36 Lattice Riesz 4-Dimensional Laplacian Let us define alattice analog of the fractional Laplace operator of the Riesztype [2 14] which is an operator for scalar fields on the latticespace-time

Definition 7 The lattice fractional Laplace operator of theRiesz type (minusΔ)

1205722

119871for 4-dimensional Euclidean space-time

is defined by the equation

((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum

1198981sdotsdotsdot1198984=minusinfin

K+

120572(n minusm) 120593119871 (m) (68)

where the constant 119886 is 119886 = (sum4

120583=11198862

120583)1205722

and the kernelK+120572(n minusm) is defined by the equation

K+

120572(n) = 1

1205874int

120587

0

1198891198961 sdot sdot sdot int

120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4

120583=1119899120583e120583 and the parameter 120572 gt 0 is the order of

the lattice operator (68)

Note that the kernel (69) is connected with (47) by theequation

1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)

where n120583 = 119899120583e120583 without the sum over 120583The Fourier series transform K+

120572(k) of the kernelsK+

120572(n)

in the form

K+

120572(k) =

+infin

sum

1198991 sdotsdotsdot1198994=minusinfin

119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)

satisfies the condition

K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)

The form (69) of the kernelK+120572(n) is completely determined

by the requirement (72)The inverse relation to (71) with (72)has the form (69)

If the lattice field 120593119871(m) depends only on one variable119898120583with fixed 120583 isin 1 2 3 4 that ism = m120583 = 119898120583e120583 without thesum over 120583 then we have

(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)

The lattice fractional Laplacian (minusΔ)1205722

119871in the Riesz

form for 4-dimensional lattice space-time can be consideredas a lattice analog of the fractional Laplacian (minusΔ)

1205722

119862for

continuum Euclidean space-time R4 that is defined by (35)

37 Lattice Fractional FieldTheory Thepath integral (11) doesnot have a precise mathematical definition To give a defi-nition of the path integrals we can introduce a space-timelattice with ldquolattice constantsrdquo a120583 Every point on the latticeis then specified by four integers which are denoted by thevector n = (1198991 1198992 1198993 1198994) where the last component willdenote a lattice analog of the Euclidean time

In the path integral expression for lattice fields we shoulduse dimensionless variables only Note that by convention allvariables of the lattice theory are dimensionless variables

For lattice fractional fied theory the path-integral expres-sion of the Green functions is

⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904

119895=1119889120593119871 (n119895) (120593119871 (n1) sdot sdot sdot 120593119871 (n119904)) 119890minus119878119864[120593119871119869119871]

intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)

The structure of the path integral (74) is analogous to thatused in the statistical mechanics of lattice system

The lattice action 119878119864[120593119871 119869119871] is not unique and we canchoose the simplest one We have only the requirement thatany lattice action should reproduce the correct continuumexpression in the continuum limit 119886120583 rarr +0

The action used in the path integral (74) can be consid-ered in the forms

119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)

For lattice theory with the lattice Riesz fractional Laplacianthe action is

119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)

Using (67) we rewrite expressions (75) in the form

119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)

where the kernel 119875119899120583119898120583(2120572) is given by

119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)


Expression (78) can be used for all positive real values 120572

including positive integer values This kernel describes thespace-time lattice with long-range properties that can beinterpreted as a lattice space-time with power-law nonlocal-ity For the lattice with the nearest-neighbor interactions thekernel 119875119899120583119898120583(120572) can defined by

119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)

Note that the kernel (78) with 120572 = 2 reproduces the samecontinuum fractional field theory as (79)

Using (68) we rewrite expression (76) in the form

119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)

andK+120572(n minusm) is defined by the expression (69)

For the lattice fractional field theory we can define thegenerating functional in the form

1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)

It can be easily calculated since the multiple integral is of theGaussian type Apart from an overall constant which we willalways drop since it plays no role when computing ensembleaverages we have that

1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1

119899120583119898120583(2120572) is the inverse of the matrix (78) and

det119875(2120572) is the determinant of 119875minus1119899120583119898120583

(2120572) The inverse matrix119875minus1

119899120583119898120583(2120572) is defined by the equation

+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)

and it can be easily derived by using the momentum spacewhere 120575119899120583119898] is given by

120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)

where 11989601205832 = 120587119886120583 and the integration is restricted by theBrillouin zone 119896120583 isin [minus11989601205832 11989601205832]

Using the discrete Fourier representation one finds that119875119899120583119898120583

(2120572) is given by

119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)

Note that the integration in (86) is restricted to the Brillouinzone 119896120583 isin [minus11989601205832 11989601205832] where 120583 = 1 2 3 4 and 11989601205832 =

120587119886120583The inverse matrix is

119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)

For the action (80) the generating functional is defined bythe equation

1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)

Using the integer-order differentiation of (89) with respect tothe sources 119869119871 we can obtain the correlation functions for thelattice fractional field theoryThe2-point correlation functionis

⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)

Using the discrete Fourier representation one finds that119875nm(120572) is given by

119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890

119894(k(x(n)minusx(m)))

(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)

Here we use the notations

x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)

The inverse matrix 119875minus1nm(120572) has the form

119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)

The right-hand side of expression (94) depends on thelattice sitesn andm andon the dimensionlessmass parameter119872119871 Let us indicate this dependence explicitly by using thenotation 119866119875(nm119872119871 120572) = 119875

minus1

nm(120572) Then substituting (92)into (94) we have

119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)


We can study continuum limit of (95) in order to extractthe physical two-point correlation function ⟨120593119862(x)120593119862(y)⟩ Totake the limit 119886120583 rarr 0 we should take into account that119909120583 rarr

119899120583119886120583 and 119910120583 rarr 119898120583119886120583 In our case the continuum limit cangive the correct continuum limit

⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)

that reproduces the result for the scalar two-point functionfor fractional filed theory with continuum space-time

4 Continuum Fractional Field Theory fromLattice Theory

In this section we use the methods suggested in [18ndash20] todefine the operation that transforms a lattice field 120593119871(n) andlattice operators into a field 120593119862(x) and operators for con-tinuum space-time

The transformation of the field is following We considerthe lattice scalar field 120593119862(n) as Fourier series coefficients ofsome function 120593(k) for 119896120583 isin [minus11989601205832 11989601205832] where 120583 =

1 2 3 4 and 11989601205832 = 120587119886120583 As a next step we use thecontinuous limit 119886120583 rarr 0+(k0 rarr infin) to obtain 120593(k) Finallywe apply the inverse Fourier integral transform to obtain thecontinuum scalar field 120593119862(x) Let us give some details forthese transformations of a lattice field into a continuum field[18ndash20]

The lattice-continuum transform operationT119871rarr119862 is thecombination of the operationsFminus1 Lim andFΔ in the form

T119871rarr119862 = Fminus1

∘ Lim ∘FΔ (97)

that maps lattice field theory into the continuum field theorywhere these operations are defined by the following

(1) The Fourier series transform 120593119871(n) rarr FΔ120593119871(n) =120593(k) of the lattice scalar field 120593119871(n) is defined by

120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)

where the inverse Fourier series transform is

120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)

and 119886120583 = 21205871198960120583 is the lattice constants

From latticeto continuum

Fourier seriestransform

Limit

ℱΔ

Inverse Fourier integral

ℱminus1 ∘ Lim ∘ ℱΔ

transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)

Figure 1 Diagram of sets of operations for scalar fields

(2) The passage to the limit 120593(k) rarr Lim120593(k) = 120593(k)where we use 119886120583 rarr 0 (or 1198960120583 rarr infin) allows us toderive the function120593(k) from120593(k) By definition120593(k)is the Fourier integral transform of the continuumfield 120593119862(x) and the function 120593(119896) is the Fourier seriestransform of the lattice field 120593119871(n) where

120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)

and x(n) = 119899120583119886120583 = 21205871198991205831198960120583 rarr x Note that21205871198960120583 = 119886120583

(3) The inverse Fourier integral transform 120593(k) rarr

Fminus1120593(k) = 120593119862(x) is defined by

120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4

120583=1119896120583119909120583 and the Fourier integral

transform of the continuum scalar field 120593119862(x) is

120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)

These transformations can be represented by the diagram inFigure 1

Comparing (98)-(99) and (102)-(103) we see the existenceof a cut-off in themomentum in the lattice field theory In thetheory of the lattice fields 120593119871(n) the momentum integrationwith respect to the wave-vector components 119896120583 is restrictedby the Brillouin zones 119896 isin [minus11989601205832 11989601205832] where 1198960120583 =

2120587119886120583In the lattice 4-dimensional space-time all four com-

ponents of momenta 119896120583 are restricted by the interval 119896 isin

[minus11989601205832 11989601205832] Therefore the introduction of a lattice space-time provides a momentum cut-off of the order of the inverselattice constants 1198960120583 = 2120587119886120583

Using the lattice-continuum transform operationT119871rarr119862(95) and (96) give the expression for the continuum fractionalfield theory

⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)


Let us formulate and prove a proposition about the con-nection between the lattice fractional derivative and contin-uum fractional derivatives of noninteger orders with respectto coordinates

Proposition 8 The lattice-continuum transform operationT119871rarr119862 maps the lattice fractional derivatives

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)

where119870plusmn120572(119899 minus119898) are defined by (47) (48) into the continuum

fractional derivatives of order 120572 with respect to coordinate 119909120583by

T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)

Proof Let us multiply (105) by the expression exp(minus119894119896120583119899120583119886120583)and then sum over 119899120583 from minusinfin to +infin Then

FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)

Using (98) the right-hand side of (107) gives

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583

As a result (107) has the form

FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)

where FΔ is an operator notation for the discrete Fouriertransform

Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)

and the limit 119886120583 rarr 0 gives

+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)

As a result the limit 119886120583 rarr 0 for (109) gives

Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)

The inverse Fourier transforms of (112) have the form

Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1

∘ Lim ∘FΔ (Dminus

119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)

where we use the connection between the continuum frac-tional derivatives of the order 120572 and the correspondentFourier integrals transforms

F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)

As a result we obtain that lattice fractional derivatives aretransformed by the lattice-continuum transform operationT119871rarr119862 into continuum fractional derivatives of the Riesztype

This ends the proof

We have similar relations for other lattice fractionaldifferential operators Using this Proposition it is easy toprove that the lattice-continuum transform operationT119871rarr119862maps the lattice Laplace operators (65) (66) and (68) into thecontinuum 4-dimensional Laplacians of noninteger ordersthat are defined by (30) (31) and (35) such that we have

T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)

As a result the continuous limits of the lattice fractionalfield equations give the continuum fractional-order fieldequations for continuum space-time


5 Conclusion

In this paper an approach to formulate the fractional fieldtheory on a lattice space-time has been suggested Note thatlattice approaches to the fractional field theories were notpreviously considered A fractional-order generalization ofthe lattice field theories has not been proposed before Thesuggested approach which is suggested in this paper canbe considered from two following points of view Firstly itallows us to give lattice analogs of the fractional field theoriesSecondly it allows us to formulate fractional-order analogs ofthe lattice quantum field theories The lattice analogs of thefractional-order derivatives for fields on the lattice space-timeare suggested to formulate lattice fractional field theoriesThe space-time lattices are characterized by the long-rangeproperties of power-law type instead of the usual latticescharacterized by a nearest-neighbors presentation (or by afinite neighbor environment) usually used in lattice field the-ories We prove that continuum limit of the lattice fractionaltheory gives the theory of fractional field on continuumspace-timeThe fractional field equations which are obtainedby continuum limit contain the Riesz type derivatives onnoninteger orders with respect to space-time coordinates

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S G Samko A A Kilbas and O I Marichev FractionalInteg rals and Derivatives Theory and Applications Gordon andBreach Science New York NY USA 1993

[2] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierAmsterdam The Netherlands 2006

[3] N Laskin ldquoFractional quantummechanics and Levy path inte-gralsrdquo Physics Letters A vol 268 no 4ndash6 pp 298ndash305 2000

[4] N Laskin ldquoFractional quantum mechanicsrdquo Physical Review Evol 62 no 3 pp 3135ndash3145 2000

[5] V E Tarasov ldquoWeyl quantization of fractional derivativesrdquo Jour-nal of Mathematical Physics vol 49 no 10 Article ID 102112 6pages 2008

[6] V E Tarasov ldquoFractional Heisenberg equationrdquo Physics LettersA vol 372 no 17 pp 2984ndash2988 2008

[7] V E Tarasov ldquoFractional generalization of the quantumMarko-vian master equationrdquo Theoretical and Mathematical Physicsvol 158 no 2 pp 179ndash195 2009

[8] V E Tarasov ldquoFractional dynamics of open quantum systemsrdquoin Fractional Dynamics Recent Advances J Klafter S C Limand R Metzler Eds pp 449ndash482 World Scientific Singapore2011

[9] V E Tarasov Quantum Mechanics of Non-Hamiltonian andDissipative Systems Elsevier Science 2008

[10] G Calcagni ldquoQuantum field theory gravity and cosmology in afractal universerdquo Journal ofHigh Energy Physics vol 2010 article120 38 pages 2010

[11] G Calcagni ldquoGeometry and field theory in multi-fractionalspacetimerdquo Journal of High Energy Physics vol 2012 article 652012

[12] S C Lim ldquoFractional derivative quantum fields at positive tem-peraturerdquo Physica A vol 363 no 2 pp 269ndash281 2006

[13] S C Lim and L P Teo ldquoCasimir effect associatedwith fractionalKlein-Gordon fieldrdquo in Fractional Dynamics J Klafter S CLim and R Metzler Eds pp 483ndash506 World Science Pub-lisher Singapore 2012

[14] M Riesz ldquoLrsquointegrale de Riemann-Liouville et le problemede Cauchyrdquo Acta Mathematica vol 81 no 1 pp 1ndash222 1949(French)

[15] C G Bollini and J J Giambiagi ldquoArbitrary powers of drsquoAlem-bertians and the Huygens principlerdquo Journal of MathematicalPhysics vol 34 no 2 pp 610ndash621 1993

[16] D G Barci C G Bollini L E Oxman andM Rocca ldquoLorentz-invariant pseudo-differential wave equationsrdquo InternationalJournal ofTheoretical Physics vol 37 no 12 pp 3015ndash3030 1998

[17] R L P G doAmaral and E CMarino ldquoCanonical quantizationof theories containing fractional powers of the drsquoAlembertianoperatorrdquo Journal of Physics A Mathematical and General vol25 no 19 pp 5183ndash5200 1992

[18] V E Tarasov ldquoContinuous limit of discrete systems with long-range interactionrdquo Journal of Physics A Mathematical andGeneral vol 39 no 48 pp 14895ndash14910 2006

[19] V E Tarasov ldquoMap of discrete system into continuousrdquo Journalof Mathematical Physics vol 47 no 9 Article ID 092901 24pages 2006

[20] V E Tarasov ldquoToward lattice fractional vector calculusrdquo Journalof Physics A vol 47 no 35 Article ID 355204 2014

[21] V E Tarasov ldquoLattice model with power-law spatial dispersionfor fractional elasticityrdquoCentral European Journal of Physics vol11 no 11 pp 1580ndash1588 2013

[22] V E Tarasov ldquoFractional gradient elasticity from spatial disper-sion lawrdquo ISRN Condensed Matter Physics vol 2014 Article ID794097 13 pages 2014

[23] V E Tarasov ldquoLattice with long-range interaction of power-lawtype for fractional non-local elasticityrdquo International Journal ofSolids and Structures vol 51 no 15-16 pp 2900ndash2907 2014

[24] V E Tarasov ldquoLattice model of fractional gradient and integralelasticity long-range interaction of Grunwald-Letnikov-RiesztyperdquoMechanics of Materials vol 70 no 1 pp 106ndash114 2014

[25] V E Tarasov ldquoLarge lattice fractional Fokker-Planck equationrdquoJournal of Statistical Mechanics Theory and Experiment vol2014 Article ID P09036 2014

[26] V E Tarasov ldquoNon-linear fractional field equations weak non-linearity at power-law non-localityrdquo Nonlinear Dynamics 2014

[27] J C Collins Renormalization An Intro duction to Renormal-ization the Renormaliza tion Group and the Operator-ProductExpansion Cambridge University Press Cambridge UK 1984

[28] M Chaichian and A Demichev Path Integrals in PhysicsVolume II Quantum Field Theory Statistical Physics and otherModern Applications Institute of Physics Publishing Philadel-phia Pa USA CRC Press 2001

[29] K Huang Quarks Leptons and Gauge Fields World ScientificSingapore 2nd edition 1992

[30] V V Uchaikin Fractional Derivatives for Physicists and Engi-neers Volume I Background and Theory Nonlinear PhysicalScience Springer Berlin Germany Higher Education PressBeijing China 2012

[31] V E Tarasov ldquoNo violation of the Leibniz rule No fractionalderivativerdquo Communications in Nonlinear Science and Numeri-cal Simulation vol 18 no 11 pp 2945ndash2948 2013


[32] N Laskin ldquoFractional Schrodinger equationrdquo Physical ReviewE Statistical Nonlinear and Soft Matter Physics vol 66 no 5Article ID 056108 7 pages 2002

[33] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 1 McGraw-Hill NewYork NY USA 1953

[34] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 1 Krieeger MelbourneAustralia 1981

[35] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series Volume 1 Elementary Functions Gordon amp BreachScience Publishers New York NY USA 1986

[36] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Fields and Media SpringerNew York NY USA 2011

Submit your manuscripts athttpwwwhindawicom

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Biophysics


ThermodynamicsJournal of


In quantum field theory the path integral approach isvery important to describe processes in high energy physics[28] The path integrals are well-defined for systems with adenumerable number of degrees of freedom In field theorywe are dealing with the case of an innumerable number ofdegrees of freedom labeled by the space-time coordinates atleast To give the path integrals a precisemeaning we can dis-cretize space and time that is we can introduce a space-timelattice The introduction of a lattice space-time correspondsto a special form of regularization of the path integrals Inthe lattice field theory the momentum space integrals will becut off at a momentum of the order of the inverse lattice con-stants The lattice regularization can be considered as a natu-ral introduction of amomentum cut-offThe lattice renorma-lization procedure can be carried out for path integrals inmomentum space The first step of the procedure is regu-larization that consists in introducing a space-time latticeThis regularization allows us to give an exact definition ofthe path integral since the lattice has the denumerable num-ber of degrees of freedom Moreover the existence of themomentum cut-off is not surprising in the lattice fieldtheories In the expression of the Fourier integral for latticefields themomentum integrationwith respect towave-vectorcomponents 119896120583 (120583 = 1 2 3 4) is restricted by the Brillouinzone 119896120583 isin [minus120587119886120583 120587119886120583] where 119886120583 are the lattice constantsThe second step of the renormalization is a continualizationprocedure that removes the lattice structure by a continuumlimit where the lattice constants 119886120583 tend to zero In thisstep of the renormalization process the momentum cut-offis removed by the continuum limit

At the present time fractional-order generalization of thelattice field theories has not been suggested Lattice approachto the fractional field theories was not previously consideredIn this paper we propose a formulation of fractional fieldtheory on a lattice space-time The suggested theory can beconsidered as a lattice analog of the fractional field theoriesand as a fractional-order analog of the lattice quantum fieldtheories For simplification we consider the free scalar fieldsIt allows us to demonstrate a number of important propertiesin details The lattice analogs of the fractional-order differen-tial operators are suggestedWe prove that continuum limit ofthe suggested lattice theory gives the fractional field theorywith continuum space-time The fractional field equationscontain the Riesz type derivatives on noninteger orders withrespect to space-time coordinates In Section 2 the fractionalfield theory on continuum space-time is considered for scalarfields and fractional-order differential operators are defined

In Section 3 the fractional-order lattice differential oper-ators of noninteger orders are considered and the latticefractional field theory for lattice space-time is proposed InSection 4 the lattice-continuum transformation of the latticefractional theories is discussed A short conclusion is given inSection 5

2 Fractional Field Theory onContinuum Space-Time

21 Scalar Field in Pseudo-Euclidean Space-Time For sim-plification we consider scalar fields in the 4-dimensional

pseudo-Euclidean space-timeR413 Let us consider the classi-

cal field equation

(◻ +1198722) 120593 (119909) = 119869 (119909) (1)

where ◻ is the drsquoAlembert operator 120593(119909) is a real field and119909 isin R4

13is the space-time vector with components 119909120583 where

120583 = 0 1 2 3 This field equation follows from stationaryaction principle 120575119878[120593] = 0 where the action 119878[120593] has theform

119878 [120593] = minus1

2int1198894119909120593 (119909) (◻ +119872

2) 120593 (119909) (2)

In the quantum theory the generalized coordinates 120593(119909)and momenta (119909) become the operators Φ(119909) and Φ(119909)that satisfy the canonical commutation relations The Greenfunctions are

119866119904 (1199091 119909119904) = ⟨Ω1003816100381610038161003816119879 Φ (1199091) sdot sdot sdot Φ (119909119904)

1003816100381610038161003816 Ω⟩ (3)

where |Ω⟩ denotes the ground state ldquophysical vacuumrdquo ofthe fields and 119879 denotes the time-ordered product of theoperators Φ(119909) These Green functions have a path integralrepresentation [28] in the form

119866119904 (1199091 119909119904) =int119863120593 (120593 (1199091) sdot sdot sdot 120593 (119909119904)) 119890

119894119878[120593]

int119863120593119890119894119878[120593] (4)

where int119863120593 is the sum over all possible configurations ofthe field 120593(119909) The effects arising from quantum fluctuationsare defined by those contributions to the integral (4) thatcome from field configurations which are not solutions ofthe classical field equation (1) and hence do not lead to astationary action

22 From Pseudo-Euclidean to Euclidean Space-Time Let usconsider the analytic continuation of (4) to imaginary times(see Section 74 in [29]) such that

1199090997888rarr minus1198941199094 (5)

We will use 119909119864 to denote the Euclidean four-vector Itallows us to consider the Euclidean space-time R4 instead ofthe pseudo-Euclidean space-time R4

13

The Euclidean action 119878119864[120593119888] is obtained from the action(2) by using the three steps

(1) firstly the replacement 1199090 rarr minus1198941199094 where 1199090 appears

explicitly(2) the use of the real valued field 120593119888(119909119864) = 120593119888( 1199094)

instead of120593( 119905) where120593119888( 1199094) is not obtained from120593(119905 ) by substituting 1199094 for 119905 noting that 120593119888(119909119864) =

120593119888( 1199094) is a real field which is a function of theEuclidean variable 119909119864

(3) multiplying the resultant expression by minus119894

As a result this leads to the expression

119878119864 [120593119888] =1

2int1198894119909119864120593119888 (119909119864) (minus◻119864 +119872

2) 120593119888 (119909119864) (6)



◻119864 =

4

sum

120583=1

1205972

1205971199092119864120583

(7)


x = 119897minus1

0119909119864 119909120583 = 119897

minus1

0119909119864120583 120593119862 = 119897

minus1

0120593119888

119872119862 = 119897minus1

0119872

(8)


119878119864 [120593119862] =1

2int1198894x120593119862 (x) (minus◻119864119862 +119872

2

119862) 120593119862 (x) (9)


◻119864119862 =

4

sum

120583=1

1205972

1205971199092120583

(10)



int119863120593119890minus119878119864[120593119862]

(11)



119885 = int119863120593119862 (x) 119890minus119878119864[120593119862] (12)


119863120593119862 = prodx119889120593119862 (x) (13)



119862[120572120583 ] where 120572 is the




D+

119862[120572

120583]120593119862 (x) =

1

1198891 (119898 120572)intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572+1(Δ119898

119911120583120593) (x) 119889119911120583

(0 lt 120572 lt 119898)

(14)




(Δ119898

119911120583120593) (x120583) =

119898

sum

119899=0

(minus1)119899 119898

119899 (119898 minus 119899)120593 (x minus (

119898

2minus 119899) 119911120583e120583)

(15)


1198891 (119898 120572) =12058732

119860119898 (120572)

2120572Γ (1 + 1205722) Γ ((1 + 120572) 2) sin (1205871205722) (16)

where

119860119898 (120572) = 2

[1198982]

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)(119898

2minus 119895)

120572

(17)




119894119906)








D+119862[2119898

120583] 120593119862 (x) = (minus1)

119898 1205972119898

120593119862 (x)1205971199092119898

120583

(18)





D+

119862[1

120583] 120593119862 (x) =

120597120593119862 (x)120597119909120583

(19)




F(D+

119862[120572

120583]120593119862 (x)) (k) = 10038161003816100381610038161003816

11989612058310038161003816100381610038161003816

120572

(F120593) (k) (20)



F(Dminus

119862[120572

120583]120593119862 (x)) (k) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

(F120593) (k)

(120572 gt 0)

(21)


119862[120572120583 ] can be




1




by the equation

Dminus119862

[120572

119895] 120593119862 (x)

=1

1198891 (119898 120572 minus 1)

120597

120597119909120583

intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572 (Δ119898

119911120583120593) (x) 119889119911120583

(1 lt 120572 lt 119898 + 1)

(22)



Dminus119862

[1

120583]120593119862 (x) =

120597120593119862 (x)120597119909120583

(23)



Dminus

119862[120572

120583]120593119862 (x)

=120597

120597119909120583

intR1


(0 lt 120572 lt 1)

(24)


119877120572 (119909) =

120574minus1

1(120572) |119909|

120572minus1120572 = 2119899 + 1 119899 isin N

minus120574minus1

1(120572) |119909|

120572minus1 ln |119909| 120572 = 2119899 + 1 119899 isin N

(25)


1205741 (120572)

=

212057212058712

Γ (1205722)

Γ ((1 minus 120572) 2)120572 = 2119899 + 1


2120572minus1

12058712

Γ (120572

2) Γ (1 +

[120572 minus 1]

2) 120572 = 2119899 + 1

(26)






Dminus

119862[2119898 + 1

120583]120593119862 (x) = (minus1)

119898 1205972119898+1

120593119862 (x)1205971199092119898+1

120583

(119898 isin N)

(27)






119862[120572120583 ] for


120583







Dminus119862[2119898120583 ] andD+119862 [




1205972119898

1205971199092119898120583

= (minus1)119898D+119862 [

2119898

120583] 997888rarr (minus1)

119898D+

119862[120572

120583]

(119898 isin N 2119898 minus 1 lt 120572 lt 2119898 + 1)

1205972119898+1

1205971199092119898+1120583

= (minus1)119898Dminus119862[2119895 + 1

120583] 997888rarr (minus1)

119898Dminus

119862[120572

120583]

(119898 isin N 2119898 lt 120572 lt 2119898 + 2)

(28)




119862[120572120583 ]




Dplusmn

119862[120572

120583]Dplusmn

119862[120572

120583] = D

plusmn

119862[2120572

120583] (120572 gt 0) (29)



119864119862and ◻

2120572plusmn



◻120572120572plusmn

119864119862120593119862 (x) =

4

sum

120583=1

(Dplusmn

119862[120572

120583])

2

120593119862 (x) (30)

◻2120572plusmn

119864119862120593119871 (x) =

4

sum

120583=1

Dplusmn

119862[2120572

120583]120593119862 (x) (31)




and ◻2120572+

119864119862for


◻11minus

119864119862= ◻2+

119864119862= ◻119864 (32)

The operators ◻120572120572+119864119862

and ◻2120572minus



◻11+

119864119862= ◻119864 ◻

2minus

119864119862= ◻119864 (33)


119864119862or◻2120572+119864119862


119864119862or ◻2120572minus119864119862





((minusΔ)1205722

119862120593) (x) = F

minus1(|k|120572 (F120593) (k)) (34)



((minusΔ)1205722

119862120593119862) (x) =

1

1198894 (119898 120572)intR4

1

|z|120572+4(Δ119898

z120593119862) (z) 1198894z

(35)

where 119898 gt 120572 and (Δ119898


(Δ119898

z120593) (z) =119898

sum

119895=0

(minus1)119895 119898

119895 (119898 minus 119895)120593 (x minus 119895z) (36)


1198894 (119898 120572) =1205873119860119898 (120572)

2120572Γ (1 + 1205722) Γ (2 + 1205722) sin (1205871205722) (37)

where

119860119898 (120572) =

119898

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)119895120572 (38)


1205722

119862120593(k) =






119878(120572)

119864[120593119862 119869119862]

=1

2int1198894x120593119862 (x) (◻

2120572+

119864119862+1198722

119862) 120593119862 (x) + int119889

4x119869119862 (x) 120593119862 (x) (39)

where ◻2120572+



2+

119864119862= minus◻119864119862



(◻2120572+

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x) (40)


(◻120572120572minus

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

((minusΔ)1205722

119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

(41)




120593119862(x119904)⟩(120572)



119866(120572)

119904119862119864(x1 x119904) =


119864[120593119862119869119862]

int119863120593119862119890minus119878(120572)

119864[120593119862119869119862]

(42)



119885(120572)

0119862[119869119862] = int119863120593119862119890

minus119878(120572)

119864[120593119862119869119862] (43)


⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩(120572)

119864=

120575119904119885(120572)

0119862[119869119862]

120575119869119862 (x1) sdot sdot sdot 120575119869119862 (x119904) (44)





x (n) =4

sum

120583=1

119899120583a120583 (45)








119871[120572120583 ] such that

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(120583 = 1 2 3 4)

(46)




119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)






+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all




+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)


+

120572(119896) = |119896|

120572 (120572 gt 0) (50)


minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)


minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)




+

120572(119896) = |119896|


119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)





120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)


120572(0) = 0





120572(119896) with



by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2


119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)




119870+

1(119899) = minus


1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N


120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)




119899) = 0 for even 119899


119871[120572120583 ] are






+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





◻119864 =

4

sum

120583=1

1205972

1205971199092119864120583

(7)


x = 119897minus1

0119909119864 119909120583 = 119897

minus1

0119909119864120583 120593119862 = 119897

minus1

0120593119888

119872119862 = 119897minus1

0119872

(8)


119878119864 [120593119862] =1

2int1198894x120593119862 (x) (minus◻119864119862 +119872

2

119862) 120593119862 (x) (9)


◻119864119862 =

4

sum

120583=1

1205972

1205971199092120583

(10)



int119863120593119890minus119878119864[120593119862]

(11)



119885 = int119863120593119862 (x) 119890minus119878119864[120593119862] (12)


119863120593119862 = prodx119889120593119862 (x) (13)



119862[120572120583 ] where 120572 is the




D+

119862[120572

120583]120593119862 (x) =

1

1198891 (119898 120572)intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572+1(Δ119898

119911120583120593) (x) 119889119911120583

(0 lt 120572 lt 119898)

(14)




(Δ119898

119911120583120593) (x120583) =

119898

sum

119899=0

(minus1)119899 119898

119899 (119898 minus 119899)120593 (x minus (

119898

2minus 119899) 119911120583e120583)

(15)


1198891 (119898 120572) =12058732

119860119898 (120572)

2120572Γ (1 + 1205722) Γ ((1 + 120572) 2) sin (1205871205722) (16)

where

119860119898 (120572) = 2

[1198982]

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)(119898

2minus 119895)

120572

(17)




119894119906)








D+119862[2119898

120583] 120593119862 (x) = (minus1)

119898 1205972119898

120593119862 (x)1205971199092119898

120583

(18)





D+

119862[1

120583] 120593119862 (x) =

120597120593119862 (x)120597119909120583

(19)




F(D+

119862[120572

120583]120593119862 (x)) (k) = 10038161003816100381610038161003816

11989612058310038161003816100381610038161003816

120572

(F120593) (k) (20)



F(Dminus

119862[120572

120583]120593119862 (x)) (k) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

(F120593) (k)

(120572 gt 0)

(21)


119862[120572120583 ] can be




1




by the equation

Dminus119862

[120572

119895] 120593119862 (x)

=1

1198891 (119898 120572 minus 1)

120597

120597119909120583

intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572 (Δ119898

119911120583120593) (x) 119889119911120583

(1 lt 120572 lt 119898 + 1)

(22)



Dminus119862

[1

120583]120593119862 (x) =

120597120593119862 (x)120597119909120583

(23)



Dminus

119862[120572

120583]120593119862 (x)

=120597

120597119909120583

intR1


(0 lt 120572 lt 1)

(24)


119877120572 (119909) =

120574minus1

1(120572) |119909|

120572minus1120572 = 2119899 + 1 119899 isin N

minus120574minus1

1(120572) |119909|

120572minus1 ln |119909| 120572 = 2119899 + 1 119899 isin N

(25)


1205741 (120572)

=

212057212058712

Γ (1205722)

Γ ((1 minus 120572) 2)120572 = 2119899 + 1


2120572minus1

12058712

Γ (120572

2) Γ (1 +

[120572 minus 1]

2) 120572 = 2119899 + 1

(26)






Dminus

119862[2119898 + 1

120583]120593119862 (x) = (minus1)

119898 1205972119898+1

120593119862 (x)1205971199092119898+1

120583

(119898 isin N)

(27)






119862[120572120583 ] for


120583







Dminus119862[2119898120583 ] andD+119862 [




1205972119898

1205971199092119898120583

= (minus1)119898D+119862 [

2119898

120583] 997888rarr (minus1)

119898D+

119862[120572

120583]

(119898 isin N 2119898 minus 1 lt 120572 lt 2119898 + 1)

1205972119898+1

1205971199092119898+1120583

= (minus1)119898Dminus119862[2119895 + 1

120583] 997888rarr (minus1)

119898Dminus

119862[120572

120583]

(119898 isin N 2119898 lt 120572 lt 2119898 + 2)

(28)




119862[120572120583 ]




Dplusmn

119862[120572

120583]Dplusmn

119862[120572

120583] = D

plusmn

119862[2120572

120583] (120572 gt 0) (29)



119864119862and ◻

2120572plusmn



◻120572120572plusmn

119864119862120593119862 (x) =

4

sum

120583=1

(Dplusmn

119862[120572

120583])

2

120593119862 (x) (30)

◻2120572plusmn

119864119862120593119871 (x) =

4

sum

120583=1

Dplusmn

119862[2120572

120583]120593119862 (x) (31)




and ◻2120572+

119864119862for


◻11minus

119864119862= ◻2+

119864119862= ◻119864 (32)

The operators ◻120572120572+119864119862

and ◻2120572minus



◻11+

119864119862= ◻119864 ◻

2minus

119864119862= ◻119864 (33)


119864119862or◻2120572+119864119862


119864119862or ◻2120572minus119864119862





((minusΔ)1205722

119862120593) (x) = F

minus1(|k|120572 (F120593) (k)) (34)



((minusΔ)1205722

119862120593119862) (x) =

1

1198894 (119898 120572)intR4

1

|z|120572+4(Δ119898

z120593119862) (z) 1198894z

(35)

where 119898 gt 120572 and (Δ119898


(Δ119898

z120593) (z) =119898

sum

119895=0

(minus1)119895 119898

119895 (119898 minus 119895)120593 (x minus 119895z) (36)


1198894 (119898 120572) =1205873119860119898 (120572)

2120572Γ (1 + 1205722) Γ (2 + 1205722) sin (1205871205722) (37)

where

119860119898 (120572) =

119898

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)119895120572 (38)


1205722

119862120593(k) =






119878(120572)

119864[120593119862 119869119862]

=1

2int1198894x120593119862 (x) (◻

2120572+

119864119862+1198722

119862) 120593119862 (x) + int119889

4x119869119862 (x) 120593119862 (x) (39)

where ◻2120572+



2+

119864119862= minus◻119864119862



(◻2120572+

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x) (40)


(◻120572120572minus

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

((minusΔ)1205722

119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

(41)




120593119862(x119904)⟩(120572)



119866(120572)

119904119862119864(x1 x119904) =


119864[120593119862119869119862]

int119863120593119862119890minus119878(120572)

119864[120593119862119869119862]

(42)



119885(120572)

0119862[119869119862] = int119863120593119862119890

minus119878(120572)

119864[120593119862119869119862] (43)


⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩(120572)

119864=

120575119904119885(120572)

0119862[119869119862]

120575119869119862 (x1) sdot sdot sdot 120575119869119862 (x119904) (44)





x (n) =4

sum

120583=1

119899120583a120583 (45)








119871[120572120583 ] such that

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(120583 = 1 2 3 4)

(46)




119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)






+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all




+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)


+

120572(119896) = |119896|

120572 (120572 gt 0) (50)


minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)


minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)




+

120572(119896) = |119896|


119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)





120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)


120572(0) = 0





120572(119896) with



by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2


119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)




119870+

1(119899) = minus


1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N


120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)




119899) = 0 for even 119899


119871[120572120583 ] are






+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







D+

119862[1

120583] 120593119862 (x) =

120597120593119862 (x)120597119909120583

(19)




F(D+

119862[120572

120583]120593119862 (x)) (k) = 10038161003816100381610038161003816

11989612058310038161003816100381610038161003816

120572

(F120593) (k) (20)



F(Dminus

119862[120572

120583]120593119862 (x)) (k) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

(F120593) (k)

(120572 gt 0)

(21)


119862[120572120583 ] can be




1




by the equation

Dminus119862

[120572

119895] 120593119862 (x)

=1

1198891 (119898 120572 minus 1)

120597

120597119909120583

intR1

1

1003816100381610038161003816100381611991112058310038161003816100381610038161003816

120572 (Δ119898

119911120583120593) (x) 119889119911120583

(1 lt 120572 lt 119898 + 1)

(22)



Dminus119862

[1

120583]120593119862 (x) =

120597120593119862 (x)120597119909120583

(23)



Dminus

119862[120572

120583]120593119862 (x)

=120597

120597119909120583

intR1


(0 lt 120572 lt 1)

(24)


119877120572 (119909) =

120574minus1

1(120572) |119909|

120572minus1120572 = 2119899 + 1 119899 isin N

minus120574minus1

1(120572) |119909|

120572minus1 ln |119909| 120572 = 2119899 + 1 119899 isin N

(25)


1205741 (120572)

=

212057212058712

Γ (1205722)

Γ ((1 minus 120572) 2)120572 = 2119899 + 1


2120572minus1

12058712

Γ (120572

2) Γ (1 +

[120572 minus 1]

2) 120572 = 2119899 + 1

(26)






Dminus

119862[2119898 + 1

120583]120593119862 (x) = (minus1)

119898 1205972119898+1

120593119862 (x)1205971199092119898+1

120583

(119898 isin N)

(27)






119862[120572120583 ] for


120583







Dminus119862[2119898120583 ] andD+119862 [




1205972119898

1205971199092119898120583

= (minus1)119898D+119862 [

2119898

120583] 997888rarr (minus1)

119898D+

119862[120572

120583]

(119898 isin N 2119898 minus 1 lt 120572 lt 2119898 + 1)

1205972119898+1

1205971199092119898+1120583

= (minus1)119898Dminus119862[2119895 + 1

120583] 997888rarr (minus1)

119898Dminus

119862[120572

120583]

(119898 isin N 2119898 lt 120572 lt 2119898 + 2)

(28)




119862[120572120583 ]




Dplusmn

119862[120572

120583]Dplusmn

119862[120572

120583] = D

plusmn

119862[2120572

120583] (120572 gt 0) (29)



119864119862and ◻

2120572plusmn



◻120572120572plusmn

119864119862120593119862 (x) =

4

sum

120583=1

(Dplusmn

119862[120572

120583])

2

120593119862 (x) (30)

◻2120572plusmn

119864119862120593119871 (x) =

4

sum

120583=1

Dplusmn

119862[2120572

120583]120593119862 (x) (31)




and ◻2120572+

119864119862for


◻11minus

119864119862= ◻2+

119864119862= ◻119864 (32)

The operators ◻120572120572+119864119862

and ◻2120572minus



◻11+

119864119862= ◻119864 ◻

2minus

119864119862= ◻119864 (33)


119864119862or◻2120572+119864119862


119864119862or ◻2120572minus119864119862





((minusΔ)1205722

119862120593) (x) = F

minus1(|k|120572 (F120593) (k)) (34)



((minusΔ)1205722

119862120593119862) (x) =

1

1198894 (119898 120572)intR4

1

|z|120572+4(Δ119898

z120593119862) (z) 1198894z

(35)

where 119898 gt 120572 and (Δ119898


(Δ119898

z120593) (z) =119898

sum

119895=0

(minus1)119895 119898

119895 (119898 minus 119895)120593 (x minus 119895z) (36)


1198894 (119898 120572) =1205873119860119898 (120572)

2120572Γ (1 + 1205722) Γ (2 + 1205722) sin (1205871205722) (37)

where

119860119898 (120572) =

119898

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)119895120572 (38)


1205722

119862120593(k) =






119878(120572)

119864[120593119862 119869119862]

=1

2int1198894x120593119862 (x) (◻

2120572+

119864119862+1198722

119862) 120593119862 (x) + int119889

4x119869119862 (x) 120593119862 (x) (39)

where ◻2120572+



2+

119864119862= minus◻119864119862



(◻2120572+

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x) (40)


(◻120572120572minus

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

((minusΔ)1205722

119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

(41)




120593119862(x119904)⟩(120572)



119866(120572)

119904119862119864(x1 x119904) =


119864[120593119862119869119862]

int119863120593119862119890minus119878(120572)

119864[120593119862119869119862]

(42)



119885(120572)

0119862[119869119862] = int119863120593119862119890

minus119878(120572)

119864[120593119862119869119862] (43)


⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩(120572)

119864=

120575119904119885(120572)

0119862[119869119862]

120575119869119862 (x1) sdot sdot sdot 120575119869119862 (x119904) (44)





x (n) =4

sum

120583=1

119899120583a120583 (45)








119871[120572120583 ] such that

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(120583 = 1 2 3 4)

(46)




119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)






+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all




+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)


+

120572(119896) = |119896|

120572 (120572 gt 0) (50)


minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)


minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)




+

120572(119896) = |119896|


119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)





120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)


120572(0) = 0





120572(119896) with



by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2


119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)




119870+

1(119899) = minus


1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N


120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)




119899) = 0 for even 119899


119871[120572120583 ] are






+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







Dminus119862[2119898120583 ] andD+119862 [




1205972119898

1205971199092119898120583

= (minus1)119898D+119862 [

2119898

120583] 997888rarr (minus1)

119898D+

119862[120572

120583]

(119898 isin N 2119898 minus 1 lt 120572 lt 2119898 + 1)

1205972119898+1

1205971199092119898+1120583

= (minus1)119898Dminus119862[2119895 + 1

120583] 997888rarr (minus1)

119898Dminus

119862[120572

120583]

(119898 isin N 2119898 lt 120572 lt 2119898 + 2)

(28)




119862[120572120583 ]




Dplusmn

119862[120572

120583]Dplusmn

119862[120572

120583] = D

plusmn

119862[2120572

120583] (120572 gt 0) (29)



119864119862and ◻

2120572plusmn



◻120572120572plusmn

119864119862120593119862 (x) =

4

sum

120583=1

(Dplusmn

119862[120572

120583])

2

120593119862 (x) (30)

◻2120572plusmn

119864119862120593119871 (x) =

4

sum

120583=1

Dplusmn

119862[2120572

120583]120593119862 (x) (31)




and ◻2120572+

119864119862for


◻11minus

119864119862= ◻2+

119864119862= ◻119864 (32)

The operators ◻120572120572+119864119862

and ◻2120572minus



◻11+

119864119862= ◻119864 ◻

2minus

119864119862= ◻119864 (33)


119864119862or◻2120572+119864119862


119864119862or ◻2120572minus119864119862





((minusΔ)1205722

119862120593) (x) = F

minus1(|k|120572 (F120593) (k)) (34)



((minusΔ)1205722

119862120593119862) (x) =

1

1198894 (119898 120572)intR4

1

|z|120572+4(Δ119898

z120593119862) (z) 1198894z

(35)

where 119898 gt 120572 and (Δ119898


(Δ119898

z120593) (z) =119898

sum

119895=0

(minus1)119895 119898

119895 (119898 minus 119895)120593 (x minus 119895z) (36)


1198894 (119898 120572) =1205873119860119898 (120572)

2120572Γ (1 + 1205722) Γ (2 + 1205722) sin (1205871205722) (37)

where

119860119898 (120572) =

119898

sum

119895=0

(minus1)119895minus1 119898

119895 (119898 minus 119895)119895120572 (38)


1205722

119862120593(k) =






119878(120572)

119864[120593119862 119869119862]

=1

2int1198894x120593119862 (x) (◻

2120572+

119864119862+1198722

119862) 120593119862 (x) + int119889

4x119869119862 (x) 120593119862 (x) (39)

where ◻2120572+



2+

119864119862= minus◻119864119862



(◻2120572+

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x) (40)


(◻120572120572minus

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

((minusΔ)1205722

119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

(41)




120593119862(x119904)⟩(120572)



119866(120572)

119904119862119864(x1 x119904) =


119864[120593119862119869119862]

int119863120593119862119890minus119878(120572)

119864[120593119862119869119862]

(42)



119885(120572)

0119862[119869119862] = int119863120593119862119890

minus119878(120572)

119864[120593119862119869119862] (43)


⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩(120572)

119864=

120575119904119885(120572)

0119862[119869119862]

120575119869119862 (x1) sdot sdot sdot 120575119869119862 (x119904) (44)





x (n) =4

sum

120583=1

119899120583a120583 (45)








119871[120572120583 ] such that

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(120583 = 1 2 3 4)

(46)




119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)






+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all




+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)


+

120572(119896) = |119896|

120572 (120572 gt 0) (50)


minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)


minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)




+

120572(119896) = |119896|


119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)





120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)


120572(0) = 0





120572(119896) with



by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2


119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)




119870+

1(119899) = minus


1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N


120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)




119899) = 0 for even 119899


119871[120572120583 ] are






+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119878(120572)

119864[120593119862 119869119862]

=1

2int1198894x120593119862 (x) (◻

2120572+

119864119862+1198722

119862) 120593119862 (x) + int119889

4x119869119862 (x) 120593119862 (x) (39)

where ◻2120572+



2+

119864119862= minus◻119864119862



(◻2120572+

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x) (40)


(◻120572120572minus

119864119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

((minusΔ)1205722

119862+1198722

119862) 120593119862 (x) = 119869119862 (x)

(41)




120593119862(x119904)⟩(120572)



119866(120572)

119904119862119864(x1 x119904) =


119864[120593119862119869119862]

int119863120593119862119890minus119878(120572)

119864[120593119862119869119862]

(42)



119885(120572)

0119862[119869119862] = int119863120593119862119890

minus119878(120572)

119864[120593119862119869119862] (43)


⟨120593119862 (x1) sdot sdot sdot 120593119862 (x119904)⟩(120572)

119864=

120575119904119885(120572)

0119862[119869119862]

120575119869119862 (x1) sdot sdot sdot 120575119869119862 (x119904) (44)





x (n) =4

sum

120583=1

119899120583a120583 (45)








119871[120572120583 ] such that

(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(120583 = 1 2 3 4)

(46)




119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)






+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all




+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)


+

120572(119896) = |119896|

120572 (120572 gt 0) (50)


minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)


minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)




+

120572(119896) = |119896|


119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)





120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)


120572(0) = 0





120572(119896) with



by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2


119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)




119870+

1(119899) = minus


1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N


120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)




119899) = 0 for even 119899


119871[120572120583 ] are






+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119870+

120572(119899 minus 119898) =

120587120572

120572 + 111198652 (

120572 + 1

21

2120572 + 3

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(47)

119870minus

120572(119899 minus 119898)

= minus120587120572+1

(119899 minus 119898)

120572 + 211198652 (

120572 + 2

23

2120572 + 4

2 minus

1205872(119899 minus 119898)

2

4)

120572 gt 0

(48)






+

120572(minus119899) =

+119870+

120572(119899) and 119870

minus

120572(119899) is odd function 119870

minus

120572(minus119899) = minus119870

minus

120572(119899) for all




+

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870+

120572(119899) = 2

infin

sum

119899=1

119870+

120572(119899) cos (119896119899) + 119870

+

120572(0)

(49)


+

120572(119896) = |119896|

120572 (120572 gt 0) (50)


minus

120572(119896) =

+infin

sum

119899=minusinfin

119890minus119894119896119899

119870minus

120572(119899) = minus2119894

infin

sum

119899=1

119870minus

120572(119899) sin (119896119899) (51)


minus

120572(119896) = 119894 sgn (119896) |119896|

120572 (120572 gt 0) (52)




+

120572(119896) = |119896|


119870+

120572(119899) =

1

120587int

120587

0

119896120572 cos (119899119896) 119889119896 (120572 isin R 120572 gt 0) (53)





120572(119896) = 119894 sgn(119896)|119896|120572 in the

form

119870minus

120572(119899) = minus

1

120587int

120587

0

119896120572 sin (119899119896) 119889119896 (120572 isin R 120572 gt 0) (54)


120572(0) = 0





120572(119896) with



by the equations

119870+

120572(119899) =

[(120572minus1)2]

sum

119896=0

(minus1)119899+119896

119904120587120572minus2119896minus2

(120572 minus 2119899 minus 1)

1

1198992119896+2

+(minus1)[(120572+1)2]

119904 (2 [(120572 + 1) 2] minus 120572)

120587119899120572+1

(55)

119870minus

120572(119899) = minus

[1205722]

sum

119896=0

(minus1)119899+119896+1

119904120587120572minus2119896minus1

(120572 minus 2119899)

1

1198992119896+2


119904 (2 [1205722] minus 120572 + 1)

120587119899120572+1

(56)




119870+

1(119899) = minus


1205871198992 119870

+

2(119899) =

2(minus1)119899

1198992 (57)

where 119899 = 0 119899 isin Z and 119870+

119898(0) = 120587

119898(119898 + 1) for all 119898 isin N


120572(119899) in the form

119870minus

1(119899) =

(minus1)119899

119899 119870

minus

2(119899) =

(minus1)119899120587

119899+2 (1 minus (minus1)

119899)

1205871198993

(58)




119899) = 0 for even 119899


119871[120572120583 ] are






+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





+

120572(119896) = |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (59)

minus

120572(119896) = 119894 sgn (119896) |119896|

120572+ 119900 (|119896|

120572) (119896 997888rarr 0) (60)




120572(119899 minus 119898) for which

plusmn





119870+

120572(119899) =

(minus1)119899Γ (120572 + 1)

Γ (1205722 + 1 + 119899) Γ (1205722 + 1 minus 119899) (61)



1 For 120572 = 2119895 we have 119870+


119895 + 1 The function 119870+






119870minus

120572(119899) =

(minus1)(119899+1)2

(2 [(119899 + 1) 2] minus 119899) Γ (120572 + 1)

2120572Γ ((120572 + 119899) 2 + 1) Γ ((120572 minus 119899) 2 + 1) (62)




119870+

120572(119899) =

119860 (120572)

|119899|120572+1

(63)


120572(119896) =

+

120572(0) + |119896|

120572+


with nonzero term +




+



Dplusmn119871

[120572

120583]Dplusmn

119871[120572

120583] = D

plusmn

119871[2120572

120583] (120572 gt 0) (64)



120572120572plusmn

119864119871and ◻

2120572plusmn



◻120572120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

(Dplusmn

119871[120572

120583])

2

120593119871 (m) (65)


◻2120572plusmn

119864119871120593119871 (m) =

4

sum

120583=1

Dplusmn119871

[2120572

120583]120593119871 (m) (66)



(◻2120572plusmn

119864119871120593119871) (n) =

4

sum

120583=1

1

1198862120572120583

+infin

sum


119870plusmn

2120572(119899120583 minus 119898120583) 120593119871 (m) (67)


119864119862and ◻

2120572+



119864119862and ◻

2120572minus



119864119871or ◻2120572+119864119871




1205722



((minusΔ)1205722

119871120593119871) (n) =

1

119886120572

+infin

sum


K+

120572(n minusm) 120593119871 (m) (68)


120583=11198862

120583)1205722


K+

120572(n) = 1

1205874int

120587

0


120587

0

1198891198964(

4

sum

120583

1198962

120583)

12057224

prod

120583=1

cos (119899120583119896120583)

(69)


where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where n = sum4




1

1205874int

120587

0


120587

0

1198891198964(1198962

120583)1205722

cos (119899120583119896120583)

=120587120572

120572 + 111198652(

120572 + 1

21

2120572 + 3

2 minus

1205872(119899120583)2

4)

(70)



120572(n)

in the form

K+

120572(k) =

+infin

sum


119890minus119894sum4

120583=1119896120583119899120583K

+

120572(n) (71)


K+

120572(k) = |k|120572 = (

4

sum

120583

1198962

120583)

1205722

(120572 gt 0) (72)




(minusΔ)1205722

119871120593119871 (m120583) = D

+

119871[120572

120583]120593119871 (m) (73)


119871in the Riesz


1205722

119862for





⟨120593119871 (n1) sdot sdot sdot 120593119871 (n119904)⟩

=intprod119904


intprod119904

119894=1119889120593119871 (n119894) 119890minus119878119864[120593119871119869119871]

(74)




119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) (◻

2120572plusmn

119864119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(75)


119878119864 [120593119871 119869119871] =1

2sum

nm120593119871 (n) ((minusΔ)

1205722

119871+1198722

119871) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(76)


119878119864 [120593119871 119869119871] =1

2

4

sum

120583=1

+infin

sum

119899120583 119898120583=minusinfin

120593119871 (n) 119875119899120583119898120583 (2120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(77)


119875119899120583119898120583(2120572)

=1

1198862120572120583

1205872120572

2120572 + 111198652(

2120572 + 1

21

22120572 + 3

2 minus

1205872(119899120583 minus 119898120583)

2

4)

+1198722

119871120575119899120583 119898120583

(78)




119875119899120583119898120583(2) = minus

1

1198862120583

sum

119904120583gt0

(120575119899120583+119904120583 119898120583+ 120575119899120583minus119904120583 119898120583

minus 2120575119899120583 119898120583)

+1198722

119871120575119899120583 119898120583

(79)



119878119864 [120593119871 119869119871] =1

2

+infin

sum


120593119871 (n) 119875nm (120572) 120593119871 (m)

+ sum

m120593119871 (m) 119869119871 (m)

(80)



119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119875nm (120572) =1

119886120572K+

120572(n minusm) +

4

sum

120583=1

1198722

L120575119899120583 119898120583 (81)



1198850119871 [119869119871] = intprod

n119889120593119871 (n) 119890

minus119878119864[120593119871119869119871] (82)


1198850119871 [119869119871]

=1

radicdet119875 (2120572)exp(1

2

4

sum

120583=1

+infin

sum

119899120583119898120583=minusinfin

119869119871 (n) 119875minus1

119899120583119898120583(2120572) 119869119871 (m))

(83)

where 119875minus1





+infin

sum

119904=minusinfin

119875119899120583119904120583119875minus1

119904120583119898120583= 120575119899120583119898120583

(120583 = 1 2 3 4) (84)


120575119899120583119898120583=

1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583 (85)




119875119899120583119898120583(2120572) = F

minus1

Δ2120572 (119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

1198891198961205832120572 (119896120583) 119890119894119896120583(119899120583minus119898120583)119886120583

(86)

where

2120572 (119896120583) =10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722

119871 (87)



119875minus1

119899120583119898120583(2120572) = F

minus1

Δminus1

2120572(119896120583)

=1

1198960120583

int

+11989601205832

minus11989601205832

119889119896120583119890119894119896120583(119899120583minus119898120583)119886120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

2120572

+1198722119871

(88)


1198850119871 [119869119871] =1

radicdet119875 (120572)exp(1

2sum

nm119869119871 (n) 119875

minus1

nm (120572) 119869119871 (m))

(89)


⟨120593119871 (n) 120593119871 (m)⟩ =12057521198850119871 [119869119871]

120575119869119871 (n) 120575119869119871 (m)= 119875minus1

nm (120572) (90)


119875nm (120572) = Fminus1

Δ120572 (k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120572 (k) 119890


(91)

where 1198960120583 = 2120587119886120583 and

120572 (k) = |k|120572 +1198722

119871= (

4

sum

120583=1

1198962

120583)

1205722

+1198722

119871 (92)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (93)


119875minus1

nm (120572) = Fminus1

Δminus1

120572(k)

= (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k(120572 (k))

minus1

119890119894(k(x(n)minusx(m)))

(94)


minus1


119866119875 (nm119872119871 120572) = (

4

prod

120583=1

1

1198960120583

)

times int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

119890119894(k(x(n)minusx(m)))

1198894k

(sum4

120583=11198962120583)1205722

+1198722119871

(95)




⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






⟨120593119862 (x) 120593119862 (y)⟩119864 = lim119886120583rarr0

119866119875(

4

sum

120583=1

119909120583

119886120583

e1205834

sum

120583=1

119910120583

119886120583

e120583119872119862 120572)

(96)








∘ Lim ∘FΔ (97)



120593 (k) = FΔ 120593119871 (n) =+infin

sum

1198991 1198994=minusinfin

120593119871 (n) 119890minus119894(kx(n))

(98)


120593119871 (n) = Fminus1

Δ120593 (k)

= (

4

prod

120583=1

1

1198960120583

)int

+119896012

minus119896012

sdot sdot sdot int

+119896042

minus119896042

1198894k120593 (k) 119890119894(kx(n))

(99)


x (n) =4

sum

120583=1

119899120583a120583 (k x (n)) =4

sum

120583=1

119896120583119899120583119886120583 (100)




Limit

ℱΔ



transform ℱminus1

120593C(x)

(k) (k)a120583 rarr 0

120593L(n)



120593119871 (n) = (

4

prod

120583=1

2120587

1198960120583

)120593119862 (x (n)) (101)




120593119862 (x) =1

(2120587)4intR4

1198894k119890119894(kx)120593 (k) = F

minus1120593 (k) (102)

where (k x) = sum4



120593 (k) = intR4

1198894x119890minus119894(kx)120593119862 (x) = F 120593119862 (x) (103)







⟨120593119862 (x) 120593119862 (y)⟩119864 =1

(2120587)4intR4

1198894k 119890

119894(kxminusy)

(sum4

120583=11198962120583)1205722

+1198722119862

(104)




(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






(Dplusmn

119871[120572

120583]120593119871) (n) = 1

119886120572120583

+infin

sum


119870plusmn

120572(119899120583 minus 119898120583) 120593119871 (m) (105)



T119871997888rarr119862 (Dplusmn

119871[120572

120583]120593119871 (m)) = Dplusmn

119862[120572

120583]120593119862 (x) (106)


FΔ (Dplusmn

119871[120572

120583]120593119871 (m))

=

+infin

sum


119890minus119894119896120583119899120583119886120583 Dplusmn

119871[120572

120583]120593119871 (m)

=1

119886120583

+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

(107)


+infin

sum


+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583) 120593119871 (m)

=

+infin

sum


119890minus119894119896120583119899120583119886120583119870

plusmn

120572(119899120583 minus 119898120583)

+infin

sum


120593119871 (m)

=

+infin

sum

1198991015840120583=minusinfin

119890minus119894119896120583119899

1015840

120583119886120583119870plusmn

120572(1198991015840

120583)

times

+infin

sum


120593119871 (m) 119890minus119894119896120583119898120583119886120583 =

plusmn

120572(119896120583119886120583) 120593 (k)

(108)

where 1198991015840120583= 119899120583 minus 119898120583


FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

1

119886120572120583

plusmn

120572(119896120583119886120583) 120593 (k) (109)


Then we use

+

120572(119886120583119896120583) =

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

minus

120572(119886120583119896120583) = 119894 sgn (119896120583)

10038161003816100381610038161003816119886120583119896120583

10038161003816100381610038161003816

120572

(110)


+

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

+

120572(119896120583119886120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = lim

119886120583rarr0

1

119886120572120583

minus

120572(119896120583119886120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

(111)


Lim ∘FΔ (Dplusmn

119871[120572

120583]120593119871 (m)) =

plusmn

120572(119896120583) 120593 (k) (112)

where

+

120572(119896120583) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

minus

120572(119896120583) = 119894119896120583

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572minus1

120593 (k) = Lim120593 (k) (113)


Fminus1

∘ Lim ∘FΔ (D+

119871[120572

120583]120593119871 (m)) = D+

119862[120572

120583]120593119862 (x)

(120572 gt 0)

Fminus1


119871[120572

120583]120593119871 (m)) = Dminus

119862[120572

120583]120593119862 (x)

(120572 gt 0)

(114)


F (D+

119862[120572

120583]120593119862 (x)) =

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k)

F (Dminus

119862[120572

120583]120593119862 (x)) = 119894 sgn (119896120583)

10038161003816100381610038161003816119896120583

10038161003816100381610038161003816

120572

120593 (k) (115)


This ends the proof


T119871rarr119862 ((◻2120572plusmn

119864119871120593119871) (n)) = (◻

2120572plusmn

119864119862120593119862) (x)

T119871rarr119862 ((◻120572120572plusmn

119864119871120593119871) (n)) = (◻

120572120572plusmn

119864119862120593119862) (x)

T119871rarr119862 (((minusΔ)1205722

119871120593119871) (n)) = ((minusΔ)

1205722

119862120593119862) (x)

(116)



5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




5 Conclusion




References











































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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