A simple proof is given of a new summation formula recently added in the literature for a terminating r + 3Fr + 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 Saalschütz summation formula for a 3F2(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 2F1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller.
Kim, Y. S., Rathie, A. K., & Paris, R. B. (2015). An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications. Mathematical Methods in the Applied Sciences, 38(18), 4891-4900. https://doi.org/10.1002/mma.3408