Some multilinear generating functions

Abstract

The multiple hypergeometric generating function (1.3) below, due to H. M. Srivastava and J. P. Singhal [Acad. Roy. Belg. Bull. Cl. Sci. (5)58 (1972), 1238–1247], applies readily to deduce multilinear generating functions for thespecial Jacobi polynomialsP n (α-n,β) (x), P n (α,β-n) (x) orP n (α-n,β-n) (x), the Laguerre polynomialsL n (α) (x), thebiorthogonal polynomialsZ n α (x; k) of J.D.E. Konhauser [Pacific J. Math. 21 (1967), 303–314], and so on, and indeed also for any suitable products of these polynomials. The present paper is motivated by the need for a multiple hypergeometric generating function, analogous to (1.3), which could apply to yield multilinear generating functions for theunrestricted Jacobi polynomialsP n (α,β) (x). Several interesting generalizations of the multiple hypergeometric generating function (1.3), and of its analogue (5.6) thus obtained, are given; many of these generalizations are shown to apply also to derive multilinear generating functions for the classical Hermite polynomialsH n(x) and for their various known generalizations considered, among others, by F. Brafman [Canad. J. Math. 9 (1957), 180–187] and by H. W. Gould and A. T. Hopper [Duke Math. J. 29 (1962), 51–63]. The multilinear generating functions (1.19), (1.22), (1.23), (1.25), (1.30), (3.3), (4.1), (4.2), (4.8), (5.5), (5.6), (6.3), (6.4) and (6.6) below are believed to be new.