Some further reduction formulas for certain classes of-generalized multiple hyper geometric series arising in physical, astrophysical, and quantum chemical applications

Abstract

The multivariable hypergeometric function

$$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$

considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric function_p F _q of one variable, Appell and Kampé de Fériet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.