Extensions of Certain Classical Summation Theorems for the Series 2 F 1, 3 F 2, and 4 F 3 with Applications in Ramanujan′s Summations

Abstract

Motivated by the extension of classical Gauss′s summation theorem for the series 2 F 1 given in the literature, the authors aim at presenting the extensions of various other classical summation theorems such as those of Kummer, Gauss′s second, and Bailey for the series 2 F 1, Watson, Dixon and Whipple for the series 3 F 2, and a few other hypergeometric identities for the series 3 F 2 and 4 F 3. As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.