Reduction and summation formulae for certain classes of generalised multiple hypergeometric series arising in physical and quantum chemical applications

Abstract

A multivariable hypergeometric function considered recently by Niukkanen and Srivastava (1984, 1985) provides an interesting unification of the generalised hypergeometric function pFq of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions of n variables, and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications. As pointed out by Srivastava, the multivariable hypergeometric function is an obvious special case of the generalised Lauricella function of n variables, which was first introduced and studied by Srivastava and Daoust (1969). By employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied, many of which have not been given by Niukkanen. The author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting summation formulae for the multivariable hypergeometric function with x1= . . . =xn=1 and x1= . . . =xn=-1 are also presented.