# Implementation of summation theorems of Andrews and Gessel-Stanton

## DOI:

https://doi.org/10.31489/2023m4/95-104## Keywords:

Generalized hypergeometric function, Srivastava-Daoust double hypergeometric function, Reduction formulas, Mathematica Program## Abstract

Generalized hypergeometric functions and their natural generalizations in one and several variables appear in many mathematical problems and their applications. Solving partial differential equations encountered in many applied problems of mathematics physics is expressed in terms of such generalized hypergeometric functions. In particular, the Srivastava-Daoust double hypergeometric function (S-D function) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics. In this paper, we introduce two general double-series identities involving bounded sequences of arbitrary complex numbers employing the finite summation theorems of Gessel-Stanton and Andrews for terminating 3*F*_{2 }hypergeometric series with arguments 3/4 and 4/3, respectively. Using these double-series identities, we establish two reduction formulas for the (S-D function) with arguments *z, *3*z/*4 and *z, *−4*z/*3 expressed in terms of two generalized hypergeometric function of arguments proportional to *z*^{3 }and −*z*^{3 }respectively. All the results mentioned in the paper are verified numerically using Mathematica Program.