An alternative proof of the extended Saalschütz summation theorem for the \(_{r + 3}F_{r + 2}(1)\) series with applications (Q2795258)

In [Integral Transforms Spec. Funct. 24, No. 11, 916--921 (2013; Zbl 1286.33006)], the authors gave a proof of a summation formula for a terminating \({}_{r+3}F_{r+2}(1)\) hypergeometric series in which \(r\) pairs of numerator and denominator parameters differ by positive integers. In this paper they give an alternative proof of this extension of the Saalschütz summation formula for a \({}_3F_2(1)\) series. It is shown that a special case leads to a well-known summation formula for a \({}_4F_3(1)\) series. Furthermore, an application of the summation formula leads to two transformation formulas involving \({}_{r+2}F_{r+1}(x^2)\) series in which \(r\) pairs of numerator and denominator parameters differ by positive integers. This generalizes earlier results by Ramanujan in the case \(r=0\). Finally, another application leads to two reduction formulas for the Kampé de Fériet function, a hypergeometric function of two variables.