Extensions of certain classical summation theorems for the series ₂F₁, ₃F₂, and ₄F ₃ with applications in Ramanujan's summations

Motivated by the extension of classical Gauss's summation theorem for the series ₂F₁ 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 ₂F ₁, Watson, Dixon and Whipple for the series ₃F ₂, and a few other hypergeometric identities for the series ₃F₂ and ₄F₃. 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.