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

摘要

A simple proof is given of a new summation formula recently added in the literature for a terminating F-r+3(r+2)(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 Saalschutz summation formula for a F-3(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 F-2(1) series, extends certain reduction formulas for the Kampe de Feriet function of two variables given by Exton and Cvijovic & Miller. Copyright (C) 2015 John Wiley & Sons, Ltd.