An alternative proof of the extended Saalschütz summation theorem for the _{r + 3}F_{r + 2}(1) series with applications

Abstract

A simple proof is given of a new summation formula recently added in the literature for a terminating rC3 F rC2 .1/ hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a 3 F 2 .1/ series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde-Chu summation theorem for the 2 F 1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller.

Degenerate Miller–Paris Transformations

“…Over the last decade important new discoveries have been made in this area due, mainly, to Miller and Paris (with important contributions by Rathie, Kim and others), which culminated in the seminal paper [13]. See also related developments in [5,9,10,11,12]. These new discoveries deal with the generalized hypergeometric functions with integral parameters differences, i.e.…”