Extensions of Certain Classical Summation Theorems for the Series ₂<i>F</i> ₁, ₃<i>F</i> ₂, and ₄<i>F</i> 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.