A NEW PROOF OF THE EXTENDED SAALSCHÜTZ'S SUMMATION THEOREM FOR THE SERIES4F3AND ITS APPLICATIONS

Abstract

Abstract. Very recently, Rakha and Rathie obtained an extensionof the classical Saalschutz’s summation theorem. Here, in this pa-per, we rst give an elementary proof of the extended Saalschutz’ssummation theorem. By employing it, we next present certain ex-tenstions of Ramanujan’s result and another result involving hy-pergeometric series. The results presented in this paper are simple,interesting and (potentially) useful. 1. Introduction and preliminariesIn the theory of hypergeometric series 2 F 1 and generalized hyperge-ometric series p F q (see, e.g., [9]; see also [12]), the following classicalSaalschutz’s summation theorem (see, e.g., [9, p. 87]):(1.1) 3 F 2 n; ; ;;1 + + n;1#=( ) n ( ) n ( ) n n ;where n2N 0 := N [f0g, N denotes the set of positive integers, and , , are independent of n, plays an important role. Received May 23, 2013. Accepted June 3, 2013.2010 Mathematics Subject Classi cation. Primary 33B20, 33C20; Secondary33B15, 33C05.Key words and phrases. Hypergeometric series