Extensions of certain classical summation theorems for the series \(_2F_1\), \(_3F_2\) and with applications in Ramanujan's summations (Q623684)

Summary: Motivated by the extension of classical Gauss's summation theorem for the series \(_2F_1\) 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 \(_2F_1\), Watson, Dixon and Whipple for the series \(_3F_2\), and a few other hypergeometric identities for the series \(_3F_2\) and \(_4F_3\). 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.