Generalizations of a Formula due to Kummer with Applications

Abstract

The aim of this research paper is to obtain explicit expressions of
$${_2F_1}\ffnk{c}{\frac{1+x}{2}}{a,\:b}{\frac12(a+b\pm \ell+1)}$$
in the most general case for any $\ell=0, 1, 2, \ldots$\ .
For $\ell=0$, we have the well known, interesting and useful formula due to Kummer which was proved independently by Ramanujan. The results presented here are obtained with the help of known generalizations of Gauss's second summation theorem for the series $_2F_1(\frac12)$, which were
given recently by Rakha and Rathie [Integral Transformas Spec. Func. {\bf22} (11) (2011), 823--840]. The results are further utilized to obtain new hypergeometric identities by using beta integral
method developed by Krattenthaler $\&$ Rao [J. Comput. Appl. Math. {\bf160} (2003), 159--173]. Several interesting results due to Ramanujan, Choi, et. al. and Krattenthaler $\&$ Rao
follow special cases of our main findings.