Ajay Shukla

Department of Mathematics,Department of Mathematics,

Sardar Vallabhbhai National Institute of

Technology, Surat-395 007, India.

e-mail: ajayshukla2@rediffmail.com

5th Lecture of Father Valles lecture series,

St. Xavier’s College, Ahmedabad

August 2nd , 2014

Laguerre polynomials occur in many fields ofresearch in science, Engineering and NumericalMathematics such as, in Quantum Mechanics,Communication theory and Numerical InverseLaplace Transform. An attempt is made to discussLaplace Transform. An attempt is made to discusson Recent Development in Integral Transforms,specially in Laguerre Transforms. We discuss aboutLaguerre Transform in one variables as well as intwo variables and its properties.

Laguerre Polynomial in one VariablesLaguerre Polynomial in one Variables

Laguerre Polynomial:

The Laguerre Differential Equation can be written as

(1)2

2 (1 ) 0

d y dyx x ny

dx dx+ − + =

( )

( )

t h e n ,

0

n n 1n 2y ( x ) c 1 x x _ _2 21 2 !

−= − + −

(2)

(2) is the series solution of the differential equation (1).4

( )

( ) ( ) ( )

( )

( )( ) ( )

0

0

2 !

n n n 1 . . . . . n r 1r rc 1 x2r 0 r !

n r n ! rc 1 x2r 0 n r ! r !

− − += −∑

=

= −∑= −

� If C0 = 1 then,

(3)

is known as simple Laguerre polynomial.

� Now if the differential equation can be changed to

(4)

( ) ( ) ( )( ) ( )

rxn

0r 2r!!rn

2n!r

1xnL ∑= −

−=

2

2 (1 ) 0

d y dyx x ny

dx dxα+ + − + =

(4)

then,

(5)

is known as Generalized Laguerre polynomial.5

2 (1 ) 0x x ny

dx dxα+ + − + =

( ) ( ) ( )( ) ( )

rxn

0r r1!rnr!n1

r1

xn

L ∑= +−

+−=α

αα

( )

+

−+=

;1

;

!

111 α

α xnF

n

� Orthogonality of Laguerre Polynomials:

(6)

� Recurrence Relation of Laguerre Polynomials:

(7)

( )( ) ( )( ) ( )

( ) ( ) 1αRe,!n

nα1Γ

1αRen,m0,dxxαmLxα

nLxex0

−>++=

−>≠=∫−∞

( )( ) ( )( ) ( ) ( )( )xLnxLnxLDx nnn

ααα α 1−+−=

(8)

(9)

(10)

6

( )( ) ( )( ) ( )( )xLxDLxDL nnn

ααα11 −− −=

( )( ) ( )( )∑−

=

−=1

0

n

k

kn xLxDLαα

( )( ) ( ) ( )( ) ( ) ( )( )xLnxLxnxLn nnn

ααα αα 21 112 −− +−−−+−=

Laguerre Transform in one VariablesLaguerre Transform in one Variables

Laguerre Transform in one Variables:

� Laguerre transform can be used effectively to solve the heat

conduction problem (Debnath et al. [3]) in a semi-infinite

medium with variable thermal conductivity in the presence of a

heat source within the medium.

Definition:

� Debnath (1960) introduced the Laguerre transform of a� Debnath (1960) introduced the Laguerre transform of a

function defined in by means of the integral

(11)

where is the Laguerre polynomial of degree and

order discussed in the first Section (5), which

satisfies the ordinary differential equation given in (4).8

)(xf ∞<≤ x0

∫∞ −==0

)()()(~

)}({ dxxfxLxenfxfL n

x ααα

)(xLn

α0≥n

1−>α

� In view of the orthogonal property of the Laguerre polynomials

(12)

where is the Kronecker delta symbol and is given by

(13)

( ) mnnmnnm

x

n

ndxxLxLxe ∂∂=∂+Γ

+=∫

∞ − 1)()(0

ααααα

mn∂ n∂

( )1+Γ

+=∂ α

αn

nn

� The Inverse Laguerre transform is given by

(14)

9

n

{ } )()(~

)(~

)(0

11 xLnfnfLxf n

n

n

ααα ∑

∞

=

−− ∂==

� Example 1:

If , then

This follows directly from the definitions, (11) and (12).

� Example 2:

( ) ( )mf x L xα= mnnm xLL ∂∂=)}({ α

Examples:

� Example 2:

If and then

where result of Erdélyi et al. [4] (p 191) is used.10

1a > −axexf −=)(

( )10

)1(

)1(!

1)(}{

++

∞ +−−

+

++Γ== ∫ α

αα αn

n

n

axax

an

andxxLxeeL

� Example 3:

� Example 4:

If where s is a positive real number, then( ) 1sf x x −=

� Example 5:

11

� Example 6:

� Example 7:

� Example 8:

12

� Example 9:

13

Basic Operational Properties of LaguerreTransform in one VariablesTransform in one Variables

We obtain the Laguerre transform of derivatives of f(x) as

15

and so on for the Laguerre transforms of higher derivatives.

which is, due to Erdelyi (1954, vol. 2, p. 192)

16

Theorem 1

17

where L stands for the zero-order Laguerre transform.

Theorem 2

If exists, then( )( ) ( )L f x f nα= ɶ

where R[f(x)] is the differential operator given by

Theorem 3

(Convolution Theorem). If and

, then( )( ) ( )L f x f nα= ɶ ( )( ) ( )L g x g nα= ɶ

18

where h(x) is given by the following repeated integral

Laguerre Polynomial of Two VariablesLaguerre Polynomial of Two Variables

In 1991, Ragab [12] defined Laguerre polynomials of two

variables as,

where is the well-known Laguerre polynomial of

one variable.

( ) ( ),,nL x y

α β

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )

,

0

1 1,

! ! 1 1

rn

n r

n

r

n n y L xL x y

n r n r r

αα β α β

α β−

=

Γ + + Γ + + −=

Γ + − + Γ + +∑( ) ( )nL xα

one variable.

The following explicit representation of is given

by Ragab [12] :

20

( ) ( ),,nL x y

α β

( ) ( )( ) ( )

( )( )( ) ( )

,

20 0

1 1,

! ! 1 1!

r sn n rn n r s

n

r s s r

n y xL x y

r sn

α β α β

α β

−+

= =

+ + −=

+ +∑ ∑

Generating Functions for :

where is a confluent hypergeometric function of two

variables and is defined as

( ) ( ),,nL x y

α β

( ) ( )( ) ( )

( )[ ],

22

1 1, ; 1, 1; ,

!

n nnL x y n x y

n

α β α βψ α β

+ += − + +

2ψ

21

[ ]( )

( ) ( )2

0 0

; , ; ,! !

k n

n k

n k k n

a x ya b b x y

k n b bψ

∞ ∞+

= =

′ =′∑ ∑

Further, in view of the following definition of Bessel’s

function , for n not a negative integer

the generating function can be written as

( )nJ z

( )( )( ) ( )2

0 1

2;1 ; ,

41

n

n

zzJ z F n

n= − + −

Γ +

the generating function can be written as

22

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )

,

0

2 2

!,

1 1

1 (1 ) 2 2

n

n

n n n

t

n tL x y

xt yt e J x t J y t

α β

α βα β

α β

α β

∞

=

− −

+ +

= Γ + Γ +

∑

If in the above generating relation , we replace

the generating relation can be written

as

( ) ( ) ( )( ) ( )

( ),

2

0

! ,1 ; 1, 1; ,

1 1 1 1

ncnn

n n n

n c L x y t xt ytt c

t t

α β

ψ α βα β

∞−

=

− − = − + + + + − − ∑

( ) ( ),,nL x y

α β

by and let ,t

t cc

→ ∞

as

23

( ) ( )( )

( ),

1

2

0

! ,1 1 ;1 ,1 ; ,

1 1 1

n

n

n n

n L x y t xt ytt

t t

α ββ ψ β α β

α

∞− −

=

= − + + + + − − ∑

Laguerre Transform in Two VariablesLaguerre Transform in Two Variables

26

27

28

BASIC OPERATIONAL PROPERTIESBASIC OPERATIONAL PROPERTIES

INVERSE OF LAGUERRE TRANSFORM INVERSE OF LAGUERRE TRANSFORM

Some Properties of Laguerre Transform in two Variablestwo Variables

37

38

39

40

41

Applications:

The study of Laguerre Transform is gaining more importance

among the researchers due to its applications in the area of

Mathematical Physics. Debnath et al.[3](p 520) had applied

Laguerre Transform to solve Heat Conduction problem as well

as to solve Diffusion equation. In his paper “An Application of

Laguerre Transform on the problem of Oscillation of a very

long and heavy chain” Debnath et al have applied Laguerre

Transform to solve Mechanical Vibration problem.Transform to solve Mechanical Vibration problem.

� G. Evangelista of Switzerland developed a new method for real

time frequency warping of sounds using short-time Laguerre

Transform.

(courtesy:infoscience.epfl.ch/record/34067/files/Evangelista00

a.pdf)

42

� Mandyam et al. [8] of The University of New Mexico has

applied Discrete Laguerre Transform to the Speech Coding.

� (courtesy:

http://www.computer.org/portal/web/csdl/doi/10.1109/ACSSC.1

995.540894)

� Sumita et al.[17] of The University of Rochester applied

Bivariate Laguerre Transform for Numerical Exploration of

Bivariate Process.Bivariate Process.

� Keilson et al. [5] used Laguerre Transform as a tool for the

numerical solution of integral equations of convolution type.

� Application of Laguerre Transforms in Two Variables.

43

ReferencesReferences

References :

1. Andrews, G.E., Askey R. and Roy R.: Special Functions, Cambridge Univ.Press, 1999.

2. Bailey W.N., Generalized Hypergeometric Series, Stechert-Hafner ServiceAgency, New York and London, 1964.

3. Debnath L. and Bhatta D., Integral Transform and Their Applications,Chapman & Hall/CRC, 2007.

4. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., HigherTranscendental Functions,McGraw Hill, New York, Vol.1,1981.

5. Keilson J. and Nunn W. R., Laguerre transformation as a tool for thenumerical solution of integral equations of convolution type, Appliednumerical solution of integral equations of convolution type, AppliedMathematics and Computation, Vol. 5(4), 313-359, 1979.

6. Lebedev N. N., Special functions and their applications, Prentice – Hall, Inc.Englewood Cliffs, N. J., 1965.

7. McBride E. B., Obtaining generating functions, Springer-Verlag, New York,1971.

8. Mandyam G. , Ahmed N. and Magotra N. , Application of the DiscreteLaguerre Transform to Speech Coding, Proc. IEEE Asilomar Conf. onSignals, Systems and Computers, 1995. 45

9. Prajapati J.C., Ph. D. Thesis submitted to S.V.N.I.T.,SURAT On

Generalization of Mittag-Leffler Function and Its Applications, 2008.

10. Rahman M., Gasper G., Basic Hypergeometric Series, Cambridge Univ.

Press, 1990.

11. Rainville E. D., Special Functions, The Macmillan Company, New York 1960.

12. Saran N., Sharma S.D., Trivedi T.N., Special Functions, Pragati Prakashan,

Meerut, 2001.

13. Sharma J.M. and Gupta R.K., Special Functions, Krishna Prakashan,

Meerut, 1996.

14. Shukla A.K., Salehbhai I.A., and Prajapati J.C., On the Laguerre transform14. Shukla A.K., Salehbhai I.A., and Prajapati J.C., On the Laguerre transform

in two variables, Integral Transforms Spec. Funct., 20 (2009) 459-470.

15. Slater L. J., Generalized Hypergeometric functions, University press,

Cambridge, 1966.

16. Sneddon I. N., Special function of mathematical physics and chemistry, 3rd

edition, Longman, London, 1961.

17. Sumita U. and Kijima M., The Bivariate Laguerre Transform and Its

Applications: Numerical Exploration of Bivariate Processes, Adv. Appl. Prob.,

Vol. 17, 683-708, 1985.46