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. The aim of this paper is to establish the well-known and veryuseful classical Saalschutz’s theorem for the series 3 F 2 (1) by following at method. In addition to this, two summation formulas closelyrelated to the Saalschutz’s theorem have also been obtained. The resultsestablished in this paper are further utilized to show how one can obtaincertain known and useful hypergeometric identities for the series 3 F 2 (1)and 4 F 3 (1) already available in the literature. 1. Introduction and results requiredWe start with the following well-known and useful classical Saalschutz’stheorem [4, p. 87, Section 51] for the series 3 F 2 (1). If n is a non-negativeinteger and if a , b , c are independent of n ,(1.1) 3 F 2 [ n; a; bc; 1+ a + b c n ; 1]=( c a ) n ( c b ) n ( c ) n ( c a b ) n : As mentioned in almost all the standard books on generalized hypergeomet-ric series that this theorem can be established with the help of the followingEuler’s transformation formula [4, p. 60, Eq.(5)]. If

A NEW PROOF OF SAALSCHÜTZ'S THEOREM FOR THE SERIES3F2(1) AND ITS CONTIGUOUS RESULTS WITH APPLICATIONS

A NEW PROOF OF SAALSCHÜTZ'S THEOREM FOR THE SERIES₃F₂(1) AND ITS CONTIGUOUS RESULTS WITH APPLICATIONS