Probabilistic Derivation of Some Generating Functions for the Laguerre Polynomials

-A well-known generating function of the classical Laguerre polynomials was recently rederived probabillstically by Lee. In this paper, some other (presumably new) generating functions for the Laguerre polynomials are derived by means of probabillstic considerations. A direct (analytical) proof of each of these generating functions is also presented for the sake of completeness. © 1999 Elsevier Science Ltd. All rights reserved. Keywords--Generating functions, Laguerre polynomials, Hypergeometric function, Noncentral negative binomial distribution, Polsson distribution, Group-theoretic method, Probability mass function, Probability generating function, Bilinear generating functions. 1. I N T R O D U C T I O N , D E F I N I T I O N S , A N D P R E L I M I N A R I E S The classical Laguerre polynomials L(n (~) (x), of order cr and degree n in x, defined by L(n a) (x) := (1) k=0 are orthogonal over the interval (0, c¢) with respect to the weight function x a e-Z; in fact, we have (ef., e.g., [11) ~ (a ) > -1 ; m ,n z N0 := NU {0}; N := {1 ,2 ,3 , . . . } , The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council o.f Canada under Grant 0GP0007353. 0898-1221/99/$ see front matter. © 1999 Elsevier Science Ltd. All rights reserved. Typeset by .4,~tS-TEX PII:S0898-1221(99)00234-5 22 P.-A. LEE e t al. where dfm,, denotes the Kronecker delta. Just as the other members of the family of classical orthogonal polynomials (e.g., the Jacobi polynomials P~(~'~)(x), the Hermite polynomials Hn (x), the Gegenbauer (or ultraspherical) polynomials Cn(x), and the Chebyshev polynomials Tn(x) and Un(x) of the first and second kinds), the Laguerre polynomials can be expressed as a hypergeometric function L(~)(x) = (n + (~) 1Fl(-n; (~ + l; (3) where 1F1 is the (Kummer's) confluent hypergeometric function which corresponds to the special case u = v = 1 of the generalized hypergeometric function uFv (with u numerator and v denominator parameters). Furthermore, since [2, p. 42, equation 1.4(3)] ~+lFv [ n ' c q ' ' ' ' ' ~ ' ; ] z (4) ] e51o, = (/31)n" "(~-'~n ( z ) n X v + l F u [ 1 a l n , . . . , 1 olu n ; which follows upon reversing the order of terms in the finite sum for either side of (4), the Laguerre polynomials in (3) can also be expressed in the form L(a)(x) = ( 2F0 n , a n; ; . (5) Here, and throughout this paper, (A)n is the Pochhammer symbol defined by F ( A + n ) { 1, (n=O; A#O), (A)n:= F ~ = _ A ( A + I ) . . ( A + n 1 ) , (ne51). (6) Recently, Lee [3] gave a probabilistic derivation of the following generating function for the Laguerre polynomials: y n ¥ L([-1)(xn=0 (7) v > 0; x, y arbitrary. In his derivation of (7), the index parameter v + t of the Negative Binomial (NB) distribution is considered with t assumed to vary as a Poisson random variable (r.v.) T with parameter A1 + A2. This turns out to be the NB mixture formulation of the Noncentral Negative Binomial (NNB) distribution (cf. [4]) with the Poisson as mixing distribution. Let X I t (X conditional on t) be a NB r.v. with parameters p and v + t, where t is a Poisson r.v. T with parameter A. Then the unconditional r.v. X has the NNB probability mass function (p.m.f.) ~(k) := Prob(X = k) = e-XPp k q~' L~-I)( -Aq) , (8) kEN0; v,A > O; 0 < p < l ; q = l p . Furthermore, the probability generating function (p.g.f.) is given by (1--qpz) V ( { q 1} ) G(z)= 79(k) z k = exp A 1 pz ' k--0 (9) Izl < The generating function (7) is a well-known (rather classical) result (cf., e.g., [5, p. 348, equation (27); 6, p. 142, equation (18); 7, p. 319, entry 48.19.2; 2, p. 172, Problem 22(ii)]). In fact, since 0n ( t ,, "l l , , Oy n Laguerre Polynomials 23 the generating function (7) is an immediate consequence of the Taylor expansion of L ( [ 1 ) ( x y) in powers of y. The work of Lee [3] stems essentially from a paper of Chatterjea [8] in which the generating function (7) was derived by applying the familiar group-theoretic method of Louis Weisner (1899-1988), which is described and illustrated fairly adequately in the works of Miller [9], McBride [10], and Srivastava and Manocha [2, Chapter 6]. On the other hand, Hubbell and Srivastava [11] (and, subsequently, Rassias and Srivastava [12]) considered a number of applications of the generating function (7) in the theory of bilinear, bilateral, and mixed multilateral generating functions for the Laguerre polynomials. The application of probabilistic methods in the theory of special functions is an interesting topic which has been discussed, among others, by Lyusternik [13] and Dickey [14]. Probabilistic derivations of various formulas and identities for special functions are sometimes elementary in comparison with the use of some other mathematical techniques. More importantly, such a derivation may be easily generalized to obtain possibly new identities. In this paper, we shall first consider an alternative probabilistic derivation of the generating function (7) based upon the elementary technique of p.g.f. This technique is then used to derive a bilinear generating function for the Laguerre polynomials. A special case of this bilinear generating function is a well-known generating function, namely (cf. [6, p. 138, equation (11); 2, p. 132, equation 2.5(5)]) c o ( sz) (~/)k L£")(z)(-s) k = (1 +s ) -~IF1 ~; # + 1; ~ . (11) ( , + 1)k k = 0 In our present investigation, we shall denote by X ,-~ NNB(u, A, p) a NNB r.v. with parameters (v, A,p) as shown. 2. DERIVATION OF (7) BY PROBABILITY GENERATING FUNCTION TECHNIQUES Let X I t be a NNB r.v. with parameters (u + t, A1, p), where t is a Poisson r.v. T with parameter A2. In this case, the p.g.f, of X I t is given by Gx]t(z) = exp A1 1 p z The unconditional p.g.f, of X is o o Gx(z) = Gxlt (z)e t~ t~O = ( 1 ~ ) ~ e x p ( A I { i qpz 1 } ) e x p ( A 2 { 1 qpz 1} ) (13, ( { ' 1}) = q exp ()~1+A2) 1--pz ' since Observe that (13) is the NNB p.g.f, with parameters (v, .Xl + A2, p). If the p.g.f.s are replaced by their respective p.m.f.s in (13), we obtain ( ~ e-~ , ,pk qV+n L(~+n-1) (-)~1 q) e -~" A~n__f = e-(;~'+~')'Pk q~ L (~-1) ( (A1 + A2) q), (15) n----0