IL NUOVO CIMENTO VOL. 107 B, N. 1 Gennaio 1992 On Generating Functions (*). E. CAPELAS DE OLIVEIRA(**) Dipartimento di Matematica, Universitd di Perugia - 06100 Perugia, Italia (ricevuto il 3 Gennaio 1991; approvato il 13 Marzo 1991) Summary. -- We present an algebraic method to obtain expansions and generating functions for the product of two hypergeometric functions. Applications are dis- cussed. PACS 02.30 - Function theory, analysis. 1. - Introduction. Some time ago Brafman [1] published a series of papers involving hypergeometric functions and confluent hypergeometric functions. Using algebraic methods Brafman discussed generating functions and expansions for the hypergeometric functions and confluent hypergeometric functions. At the same time Ossicini [2] and Toscano [3] published papers involving products of ultraspherical polynomials and product of Laguerre polynomials with Jacobi polynomials. In the sixties Miller [4] published a book involving special functions and group theory where he discussed many addition theorems, many generating functions and many sum rules. A few years Bessel functions. special functions We note that ago Ciocci et al. [5] obtained expansions and sum rules involving Montaldi and Zucchelli6] rederived a series of sum rules involving without solving the respective differential equations. the methodology of group theory and the algebraic methods involve only functions of the fwst kind, i.e. the first linearly independent solution of the re- spective differential equation. Recently [7] we discussed sum rules for the product of two functions of different kinds using the methodology of the Green's function. This methodology is a global methodology because it permits to obtain sum rules for the special functions and for the polynomials. (*) The author of this paper has agreed to not receive the proofs for correction. (**) Permanent address: Departamento de Matem~tica Aplicada, IMECC-UNICAMP, 13083 Campinas, SP, Brasil. 59