Generating functions for Hermite polynomials of arbitrary order

5

January 1998 PHYSICS

LETTERS

A

Physics Letters A 237 ( 1998) 189-191

Generating functions for Hermite polynomials of arbitrary order EM. Ferniindez I CEQUINOR.

Facultad

de Ciencias

Exactas,

Universidad

National

de La Plats.

Calle 47 y 115, Casillu

de Correo

962.

1900 Lu Plata, Argentina

Received 3 July 1997; revised manuscript

received 27 August 1997; accepted Communicated by J.P. Vigier

for publication

27 October

1997

Abstract We propose a systematic method for the construction of generating functions for Hermite polynomials of arbitrary order. The procedure is based on a suitable formula for the Hermite polynomials and our results contain ones obtained earlier by Nieto and Truax [Phys. Lett. A 208 ( 1995) 81 as particular cases. @ 1998 Elsevier Science B.V.

1. Introduction In a recent Letter, Nieto and Truax [ I] developed and discussed generating functions for Hermite polynomials of arbitrary order which appear in the study of coherent and squeezed states. Their method is based on the Rodrigues formula for the Hermite polynomials (see Ref. [ 21, pp. 4647). The purpose of this Letter is to call attention to an alternative expression for the Hermite polynomials H,,(x) that is not well known, although it proves more suitable than the Rodrigues formula for some calculations. It reads (see Ref. [ 21, pp. 46, 47)

In order to derive Eq. ( 1) rewrite the well-known generating function for the Hermite polynomials (see Ref. [ 21, pp. 46,47) G(Z,X)

= exp(2zx

-

z*) =

2 $ff,(x)

(2)

n=a . as G(z,x)

= e(D)

exp(2zx)

= 2 $Z”@(D)x”, n-Q . (3)

and compare Eqs. (2) and (3). H,,(x)

= 29’(D)x”,

W(D)

D=$

= exp(-0*/4),

(1)

and to obtain H,,(x) simply expand l%‘(D) as 1 D2/4 + D4/32 + . . .; the series terminates because D2k.xA= 0 if 2k > n.

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2. More general generating functions There are expressions similar to Eq. ( 1) which are useful to obtain generating functions and recurrence relations for other polynomials (see Ref. [ 21, pp. 46, 47, and Ch. 5). In what follows we show that Eq. ( 1) is most useful for the construction of alternative generating functions for the Hermite polynomials.

EM. Ferndndez/Physics

190

Given a function f(x) desired coefficients

with a Taylor series with the

Letters A 237 (1998) 189-191

G(z,x)

1 =

+ we define the generating

function

G(z,x)

= ~c.z”H.(x). II=0

= @(D)f(2zx)

(5)

= 2 5 Cajby, n=o . j

j and obtain the generating

G(z,x)= CajeXp(2zbjx A particular

case discussed

Wkx)

fc

O”

=

c

=

]exp(2~m/.W

(7)

Xjn+k

(8)

n=O (jn + k) ! *

Another class of generating functions ear combinations of Gaussians,

f(x)=Caje~p(bjx~) j

=F$xnjby. n=O '

= I~‘(D)(~zx)~~(D)-‘G(z,~) = (~z)~(x

- D/2)kG(z,x).

(12)

= exp( x2) we have

= J&-$w

(13)

and Eq. (12) gives us G(Lz,x)

2zx = (1 +4z2)3,2e~~

Nieto and Truax [ 11 added these two generating functions to obtain a series in which the coefficients of the Hermite polynomials are given by the coefficients of the Taylor expansion for ( 1 + x) exp ( x2). According to the discussion above, one easily obtains analytic expressions for G(z,x) from arbitrary functions of the form f(X)

(9) j

(10) example consider

= ~(D)(2~x)~f(2zx)

arises from lin-

In this case we have

As a particular

exp(-s)

For example, for f(x)

by Nieto and Truax [ 1] is

nt=,

G(k,z,x)

G(zJ)

- z’bj).

exp[xexp(2nim/j)]

1 2J_

In order to enlarge the classes of generating functions mentioned above, notice that if we have the generating function G( z, x) for f(x), we easily obtain the generating function G( k, z, x) for xkf( x) by means of the following expression (Ref. [ 21, pp. 46, 47))

(6)

function

j

2J_

(11)

The problem reduces to obtaining a closed-form expression for l%(D) f( 2zx), which is straightforward if one selects f(x) conveniently, as illustrated below. Because the exponential function exp(bx) is an eigenfunction of D with eigenvalue b, we choose

f(x) = Cajexp(bjx)

=ti(D)cosh[(2z~)~]

= CUjXmJ

exp(bjx +

(15)

djX2)

for arbitrary real numbers aj, bj, and dj, and positive integers mj. These adjustable parameters give one great freedom to build the generating functions. The formula ( 1) is also useful for the construction of generating functions for products of Hermite polynomials. They are simply given by G(z,.r,y)

= exp(-D:/4

which generalizes Truax [ 11.

the

- D;/4)f(4zxy), expression

of

(16) Nieto

and

fi M. Ferndndez/Physics

3. Conclusions We have proposed a systematic method for the construction of generating functions for Hermite polynomials of arbitrary order. It is based on Eq. ( 1) which appears to be more convenient than the Rodrigues formula for this application. The procedure consists of two steps: First, obtain a function f(x) with the desired Taylor coefficients; second, apply the exponential operator a(D) to f( 2~) (f(4zny) in the case of products of Hermite polynomials). The general results obtained above contain those given by Nieto and Truax [ 1] as particular examples. However, it has not been our main purpose to show new results but to call attention to Eq. ( I), which greatly facilitates the calculation and may be useful

Letters A 237 (I 998) 189-191

191

in other physical applications. Our method makes it clear why the generating functions for the Hermite polynomials are commonly based on exponential and Gaussian functions [ 11. The reason is that the application of the operator q(D) to such functions produces simple analytic expressions. It is worth mentioning again that there are formulae like Eq. ( 1) for other polynomials commonly encountered in many physical contexts (see Ref. [ 21, Ch. 5).

References [ 1 I M.M. Nieto, D.R. Ttuax, Phys. Lett. A 208 (1995) 8. [2] EM. FemBndez, E.A. Castro, Algebraic Methods in Quantum Chemistry and Physics (CRC, Boca Raton. 1996).