Some bilateral generating functions for a certain class of special functions. I

AbstractThe object of this paper is to present a systematic introduction to and several interesting applications of a general method of obtaining bilinear, bilateral or mixed multilateral generating functions for a fairly wide variety of special functions in one, two and more variables. The main results, contained in Theorems 2 and 3 below, are shown to apply not only to the Bessel polynomials, the classical orthogonal polynomials including, for example, Hermite, Jacobi (and, of course, Gegenbauer, Legendre, and Tchebycheff), and Laguerre polynomials, and to their various generalizations studied in recent years, but indeed also to such other special functions as the Bessel functions, a class of generalized hypergeometric functions, the Lauricella polynomials in several variables, and the familiar Lagrange polynomials which arise in certain problems in statistics. It is also indicated how these general results are related to a number of known results scattered in the literature