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Analytic solutions of the heat equation and someformulas for laguerre and hermite polynomialsDavid Colton a & Jet Wimp ba Department of Mathematical Sciences, University of Delaware, Newark, Delawareb Department of Mathematics, Drexel University, Philadelphia, PennsylvaniaPublished online: 29 May 2007.

To cite this article: David Colton & Jet Wimp (1984) Analytic solutions of the heat equation and some formulas forlaguerre and hermite polynomials , Complex Variables, Theory and Application: An International Journal: An InternationalJournal, 3:4, 397-412

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<',,nplt.\ I 't1m1hIt~. 1984. VoI 3. pp. 397 -4 12 0278-1077/R4/0304 0397 $18.50/0 ,<' Gordon and F3reaL.h. Science Publarhcr\. Inc.. I984 Prlntrd In l in~ ted Stater ~ v f Arncr~ca

Analytic Solutions of the Heat Equation and Some Formulas for Laguerre and Hermite Polynomials*

DAVID COLTON

Department of Mathematical Sciences, University of Delaware, Newark, Delaware

and

JET WIMP

Department of Mathematics, Drexel University, Philadelphia, Pennsylvania

(Received February 20, 1984 )

We consider analytic solutions of the heat equation u, , + u,., = u, defined in a cylinder and show that any such solution can be expanded in a series of polynomial solutions to the heat equation. If we define the independent complex variables z and z by r = x + (v , 2 = .r - I,,. where .r and .I. are ~ndependent complex variables, it is shown that any real-valued analytic solution of the heat equation is uniquely determined by its values on 2 = 0 or r = 0. Using this result, and expressing the above mentioned polynomial solutions to the heat equation in terms of Laguerre polynomials, we obtain some generating functions for Laguerre polynomials, as well as connection formulas between products of Hermite polynomials and Laguerre polynomials of argument r 2 = x 2 + ,y2. These connection formulas generalize a well known result of Feldheim.

AMS (MOS): 35K05, 33A65

I. INTRODUCTION

Analytic solutions of the heat equation exhibit behavior that is not shared by classical, but nonanalytic. solutions. For example analytic

*The research of the first author was supported in part by AFOSR Grant 81-0103 and the second author was supported in part by NSF Grant MCS-8301842.

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398 D. COLTON AND J . W I M P

solutions of the heat equation in one space variable can be uniquely continued into an infinite strip parallel to the x axis regardless of the boundary data satisfied by such solutions. whereas classical, nonana- lytic solutions may have natural boundaries through which no contin- uation is possible (cf. [6]). In this paper we shall be concerned with

. . p~!ync?mia! expansicxs t.f axa!ytic sc!utions of the heat equat:c;n in two space variables. In agreement with the above opening comments, the expansion theorems we shall obtain are not valid for nonanalytic solutions. We shall apply our results tn nhtain w m e genrra!ing functions for Laguerre polynomials, as well as new connection formu- lae between Hermite and Laguerre polynomials.

The problem of obtaining polynomial expansions of analytic solu- tions of the heat equation has previously been investigated by Widder ([7]). In the case of one space variable. Widder showed that if u is an analytic solution of the heat equation

where the series is absolutely and uniformly convergent on compact subsets of - co < x < m, 1 1 1 < to, and the h, are the heat polynomials defined by.

where H , denotes the Hermite polynomial of degree n (for the purpose of motivation, we shall give a new short proof of this result of Widder in Section I1 of this paper). Widder was also able to extend this result to the case of two space variables, i.e., if u is an analytic solution of the heat equation

for - LC < x < LC, - co < y < co, l t l < I, buch that u ( n , y.0) ib an entire function of prescribed growth then u can be represented in the

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ANALYTIC SOLUTIONS OF 'THE HEAT EQUATION 399

form

. . where the se::es :s abso!ute!y zr~c! mifor!x!y convergen! on compact subsets of - m <. x < m. i r i < r,. - ~ / 3 < <: w ([GI). Note in partic- ular that the representation (1.5) is only valid for a subclass of sol~utions of the heat equation that are analytic in the infinite slab -cc < x < co, -co < y < co. It1 < to. In particular, in contrast to the case of one space variable, analytic solutions of the heat equation in the cylinder x 2 + y2 < a*, It1 < to. cannot in general be extended to an infinite slab such that (1.5) is valid.

In Section 111 of this paper we shall obtain an expansion theorem for analytic solutions of the heat equation defined in the cylinder x2 + v 2 ( a2, It1 < r , where a and t , are positive constants. As in the case of one space variabie (c.i. Section i i ) our proof is baseu on I he method of Integra! operators for the heat eyuatiw. Since ihe region o f convergence of the series expansion (i .5) is ari iiifinite slab, we shall, of course, need 2 different basis !han the he.! piv!?!,rni.:ii?; -, - - - - - - h,,,(r; z) hn(; t , t ) . This new basis, surprisingiy enough, is a i m a set of polyno- mial solutions to the heat equation, in this case defined by

- - (rn rn!n! + n)! t m r n c o s n ~ ~ : ( - $ j

-, \

- - m!n! rmrnsinnsl:(- 5) ( m + n)!

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400 D. COLTON AND J. WIMP

where (r. 0) are polar coordinates and Lf denotes a Laguerre polyno- mial of order m and index n. We shall show in Section 111 of this paper that if u is a real valued analytic solution of the heat equation in the (complex) cylinder x2 + v 2 < a2, It1 < to, then u can be repre- sented in the form

where the series is absolutely and uniformly convergent on compact subsets of x 2 + Y 2 < a2, It1 < to. The expansion (1.7) can be viewed as the analogue for the heat equation of Taylor's theorem for analytic functions of a complex variable. We shall show that the representa- tion (1.7) implies that analytic solutions of the heat equation (1.4) are uniquely determined by their values on the plane t = 0, a result which is not true for nonanalytic solutions of the heat equation. This provides a way of determining the coefficients in ( i . 5 ) and r i . 7 ~ .

In the final section of this paper. we sha!l use the above resu!ts !o obtain generating functions of the form

for Laguerre polynomials having equal order and index, as well as explicit connection formulae between the product of Hermite polyno- mials Hn(x)H, , , (y ) and the Laguerre polynomial L ~ ( x ~ + .Y2 ) . This last result generalizes and provides an inverse relation for an old formula of Feldheim (c.f. 131, p. 195). These applications to special functions of our results on the heat equation are presented merely by way of example, and we leave to the reader the further derivation of new results on Hermite and Laguerre polynomials that are possible through the use of the methods contained in this paper.

II. ANALYTIC SOLUTIONS OF THE HEAT EQUATION IN ONE SPACE VARIABLE

In this section, we shall provide a new proof of Widder's result that every analytic solution of the heat equation in one space variable can be expanded in a series of heat polynomials hn(x. t ) as defined by (1.3). Our proof is based on the method of integral operators and is presented here to motivate a similar analysis for the heat equation in

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ANALYTIC SOLUTIONS O F T H E HEAT EQUATION 40 1

two space variables that will be presented in the next section of this paper. We note in passing that classical solutions of the heat equation are always analytic with respect to the space variables (c.f. 141).

THEOREM I Let u he a solurion of rhe hemr equtrtiun ( 1 . 1 ) defined it7 - - - u u, < u < x,. ti < I,. such that for each Jixed x, - s, < x < x,. u ( x , I ) is an anal~~t ic . . function of t in the disk It/ < r,. Then u can he represented in rhe form

where the h, are defined hi, (1 .3 ) and rhe cerier is ah.rolure~v and uniform!^ convergent on cotnpacl subsets of - x, < .r < x, 1 ti < t,.

Remurk The theorem implies that every analyt~c solution of the heat equation defined o n a rectangle can be analhtically continued into an infinite strip parallel to the x axls.

Proof - nf - Thenrcrn R! Holmgen's uniqwncss theorem and d~recr ca!cu!at:on we can rqresent u in the form

where 6 > 0 and

i.e., the right hand side of (2.1) 1s a solution of the heat equation f o r - x, < x < x,, Iti < to, having the same Cauchy data as u ( x , t ) , and hence equals u ( x , t ) as stated. But E is analytic for 1x1 < co, I tl > 0. and

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402 D. COLTON AND J. WIMP

Hence, if we expand u(0, t ) and u,(O,t) in a Taylor series valid for It1 < to and integrate termwise, we arrive at the desired series repre- sentation for u.

Ill. ANALYTIC SOLUTIONS OF THE HEAT EQUATION IN TWO SPACE VARIABLES

We shall now generalize the results of the previous section to the case af the heat equat im i:: t:: '~ space va:iab!es. 1:: mde: te d~ this, we shall need an integral operator for the heat equation in two space variables that is analogous to the operator (2.1) for the heat equation in one space variable. This operator has already been constructed in [2]. In particular, let u be a real valued (for x , y , t real) solution of the heat equation (1.4) defined in x 2 + y2 < a2, It1 < to. such that for each fixed ( x , y ) in the disk .u2 + y2 < a2, u ( x , y , t ) is an analytic function of t in the disk It1 < to. Let z = .w + iy. F = x - iy. Then from [2] we see that there exists a function f(z, t ) analytic for lzl < a, [ti c.: i,, such that U(: , 5, t ) = u ( z , J , t ) has the rcpresentativn

where r2 = zr, 6 > 0 and

the representation (3.1) being valid for x 2 + y2 < a, It1 < to. From (3.1) we can immediately deduce the following theorem, which will be useful for us in connection with our applications to special functions:

THEOREM 2 Let U(z,F, t ) = u ( x , y , t ) be a real valued solution of the heat equation (1.4) defined in x 2 + y 2 < a2, It1 < to, such that for each fixed ( x , y ) in the disk x 2 + y2 < a2, u ( x , y , t ) is an analytic function of t in the disk It1 < to. Then u is uniquelv determined by the function U(z. 0. t). where z and 2 are considered as independent complex vari- ables (i.e., x and y are independent complex variables).

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ANALYTIC SOLUTIONS OF THE HEAT EQUATION 403

Proof From (3.1) and (3.2) we have

where f(z. t ) = I(?. r ) and t is kept real. Hence, if

then

From (3.5) we see that if U ( z , 0, r ) is known then f ( z , t ) is determined up t o ii purciy ci;rr;piex cunstdn: (depctiding on 1') and f:om (7 ! ) ::re can now uniquely determine l / ( z , j. r ) .

x- - \.l/e are now in a position io prove iiui dcsired expansion thesrex for analytic solutions of the heat equation in :wo space variables.

THEOREM 3 Let u be a real valued solution of the heat equation (1.4) satis-ving the hypothesis of Theorem 2. Then u(x , y, t ) = u ( r , 0 , t ) can be represented in the form

where the functions v i , and c:, are defined in (1.6) and the series is absolurely and uniformly convergent on compact subsets of x 2 + -v2 < a2,

Ill < to.

Proof Our proof is modelled after that of Theorem I for the heat equation in one space variable. We simply use the representation (3.1) and expand f in its Taylor series

which is absolutely and uniformly convergent for IzI < a , It1 < I,. The

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404 D. COLTON AND J. WIMP

theorem now follows by substituting (3.6) into (3.1), integrating termwise. and using the result that

The following theorem will be used in our forthcoming section on special functions and is an immediate consequence of Theorem 3.

THEOREM 4 Let u be a real valued solution of the heat equation (1.4) satisfying the h-ypothesis of Theorem 2. Then u is unique!^ determined bv its vulues on t = 0.

Proof Fiurn Tlic~irem 3 we have that

Y. m i n! ,.2n1 + rr

~ ( r - 8 . 0 ) = 2 C (amn 4mm(m + n)! cos no n=Om=U

n! r2m + n

+ bmn ~ ( m + n ) ! sin n 8 ) (3.8)

and hence by orthogonality

00

f: ')(r) = rn C am,, n! rZm m = o 4"(m + n) !

and

00

n! f: " ( r ) = rn x bmn rZm

m = ~ 4"(m + n)!

are known. Since f(z,t) is analytic for lzl < a, It1 < to, we can conclude that the series in (3.9) are analytic functions of r in some neighborhood of the origin and hence the coefficients am, and bmn are uniquely determined.

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ANALYTIC SOLUTIONS O F THE HEAT EQUATION 405

IV. GENERATING FUNCTIONS AND CONNECTION FORMULAS FOR LAGUERRE POLYNOMIALS

We shall now show how the results of the previous section can be used to obtain generating functions for Laguerre polynomials as well as connection formulae between Hermite and Laguerre polynomials. We first turn our attention to obtaining generating functions through the use of the integral operator (3.1). (3.2). T o this end. we first set f ( z . t ) = e" in (3 .1 ) and define H ( z , 2. r ) by

Computing the residue in ( 4 . 1 ) yields

and from the integral representation

- - T(k + 1/2)*

, F , ( k + 1/2,2k -t- I, - z t ) (2k)!

where , F, denotes a hypergeometric function and I, a modified Bessel function, we have from (4.2) and Legendre's duplication formula that

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406 D. COLTON AND J. WIMP

From [3], p. 100, equation (12) we have

and hence from (4.4), (4.5) we have

where r = z2. On the other hand, if we return to (4.i j, expand the exponential in its Taylor series, and use (3.7) and Legendre's duplica- tion formula, we have

Hence, from (4.6) and (4.7) we see

Considering z and 2 as independent complex variables and setting t = 1, z = 4 x , = - y now gives the generating function

The generating function (4.9) can be put in normal form by replacing

Y b y y / ' x . A different (known) generating function can be found by substitut-

ing f(z, t ) = ( 1 - zt)- ' in (3.1) and defining H(z.2 , t ) b y

k - 1 / 2 ( r2s2z lk(] - s2)

ds; I z t l< l l k - 0 4 * ~ ( k + 1/2)(1 - z t (1 - s ~ ) ) ~ + '

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ANALYTIC SOLUTIONS O F THE HEAT EQUATlON 407

From the identity

- - T ( k + 1 /2)'

F(2k + 1 ) ,F , (k + 1 , k + 1 / 2 , 2 k + 1; t z )

where - F 1 denotes a hypergeometric functionl we see from (411)). (4.1 I ) and Legendre's duplication formula that

On the other hand, we have as in (4.7) that

and hence setting t = 1, z = 4x, t = -,v and letting?? j y / x gives the generating function

r 1

(4.14)

This is contained in a result of Carlitz [ I ] .

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408 D. COLTON AND J . W I M P

We note in passing that by an appropriate choice of f ( i . 1 ) in ( 3 . 1 ) we can obtain most of the well known generating functions for Laguerre polynomials of integer index, for example, those listed in [S].

It can be determined by the Fasenmyer method, as formulated by Zeilberger [8], that the poljnomial J;,(.Y) = L,y( u) which occurs in :hese expar,sior,s satisfies a recurrence re!zt~on :!f erder :I! !e:ist :wo with coefficients which are polynomials in .u. The degree of one o f thesc polynomials is at least three. Determining the coefficients is a very tedioris process which involves wlving I ! e q ~ ~ l t i o n s in I I un- knowns whose coefficients are polynomials in n. The result is, how- ever. rather simple. We find:

This is t j i e f i r s t i n 5 t a n e e knew c,f ...xjhere 8 reciirreiice

-- i C l a L i t , l l I-+:,... f ~ i 2 polynomial of hypergeometric type has c~efficients which are p~!yncmia!s . - ~f s w h high degree.

We conclude this section by deriving connection formulas between the orthogonal polynomials in two variables H,, (x) H,,, (,v) and L ~ ( X > + y2) , thus generalizing a well known result of Feldheim. For the sake of simplicity. we assume that t1 = 2p is an even integer and consider the polynomial solution of the heat equation defined by

m ! n ! c,:,, ( r , 6, t ) = tm t "cos nHL,:( - r2/4r) . (4.1 5 )

( t 7 1 + ? I ) !

At t = 0 we have

&,(r , 6.0)

- - n! ,.n+Z,n cos no 4'"(m + n ) !

- - "! ( X ~ + ~ ~ ) ' ~ R ~ Z " 4"'(m + n ) !

- - n ! ' 2 C A x 2 t + + ~ m - 21

Y ' ( m + n) ! ,=,

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ANALYTIC SOLUTIONS O F THE HEAT EQUA'TION 409

where

From Theorem 4 , and the fact that h,(x,t)h,,,O,. I ) (where the h, are defined by ( 1 3)) are solutions of the heat equation ( I .4) , we have that

m! t "r2Pcos 2 p 8 ~ i p ( - r 2 / 4 t )

and setting t = - 1 / 4 gives

where x = r cos6. y = r sin 8 and the c, are defined by (4 .17) . We remark that when p = 0, c, = (p), and (4.19) reduces to Feldheim's formula [3, p. 195 (32)]. Note, however, that the formula in this reference 1s incorrect, lacking a 4" factor on the right hand side. It is true that c, can also be written as a hypergeometric series (a,F2) but (4.17) is the more convenient formula. In using (4.17) . one should remember the convention (r) = 0 when n > m.

I t can be verified in the same manner that when n = 2p + 1 one obtains

where

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410 D. COLTON AND J. WIMP

We now want to obtain the inverse connection formula to (4.18). i.e.. to expand a product of Hermite polynomials in a series of Laguerre polynomials. To this end. we again use the fact that the product of heat polynomials h,(.r, r)h,(.l?, r ) is a solution of the heat equation (1.4) and by Theorem 2 is uniquely determined by its values on 2 = 0 where z = x t iy , 2 = x - y. wiii i x aiid -L. cuiis~dei-ed as independent complex variables. Hence we evaluate

on 7 = 0 to obtain

where

From (1.6) and the identities

we see that

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ANALYTIC SOLUTIONS O F THE HEAT EQUATION 41 1

For the sake of simplicity, we assume that n = 2p and m = 2q are even. Then the solution of the heat equation defined by

where

is real for x, y, t real and agrees with h, ,(x, t ) h , ( y , t ) on Z = 0. Hence, by Theorem 2, u ( r , 6, Z) = h,(x. t)h,(j. . r ) everywhere, i.e.,

Setting f = - 1/4 now gives the desired connection formula,

where x = r cos 8, y = r cos 6 and

( - ~ ) ~ k ! (2p + 2q - 2k)! dk =

(2p + 2q - k)!

( - 1 ) ' , =o (2p - 2 j ) ! j ! (29 - 2 k + 2 j ) ! ( k - j ) !

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412 D. COLTON AND J. WlMP

References

[I] Carlitz, L.. Some generating functiuna for Laguerre polqnomials. Duke Marh. J. 35 (1968). 825-827.

[2] Colton. D.. Runge's theorem fur parabol~c equarions in rwo space car~ables. Proc. R o w 1 Soc. Edinburgh, 73A ( 1975). 307- 3 15.

[3] Erdelyi. A.. Magnus. W.. Oberhettinger. F. and Tricom~. F. G.. Hi£her Transcen- denra! Funcrions. Vol. 11, McGraw Hill, KY. !953

(41 Friedman. A,. Parrial Dlfferenriul Eyuc~rions. Holr. Rinehart and Wlnstun. N Y . 1969.

rcl ,>I D-;.....!I- L\ULL, " n L I L . E,, sPccla! ,ru,7c:;o,7s, ~-~).i;! i~:~, y ~ , 1950,

16) W~dder , D. V.. The Hear Eyuarion. Academ~c Press. NY. 1975. [7] Widder, D. V.. Analytic solurions to the heat equation. Duke Math. J. 29 (1962).

497-504. [8] Zeilberger. D.. Sister Celine's technique and its generalizations. J. Math. Anal. Appl.

85 (l982), 114-145.

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