An alternative proof of the extended Saalschutz summation theorem for the r+3Fr+2(1) series with applications

Abstract

A simple proof of a new summation formula for a terminating r+3Fr+2(1) hypergeometric series, representing an extension of Saalschutz's formula for a 3F2(1) series, is given for the case of r pairs of numeratorial and denominatorial parameters differing by positive integers. Two applications of this extended summation theorem 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 2F1 series, extends certain reduction formulas for the Kamp?e de F?eriet function of two variables given by Exton and Cvijovi?c and Miller.