### Generalization of a Transformation Formula Due to Kummer and Ramanujan With Applications

#### Abstract

The aim of this research paper is to find the explicit expressions of

\[

_{2}F_{1}\left[

\begin{array}

[c]{ccc}%

a, & b; & \\

& & \frac{1+x}{2}\\

\frac{1}{2}(a+b+i+1); & &

\end{array}

\right]

\]

for $i=0,\pm1,\ldots,\pm9.$

For $i=0$, we have the well known, interesting and useful formula due to Kummer which was independently discovered by Ramanujan. The results are derived with the help of generalizations of Gauss's second summation theorem obtained recently by Rakha et al.

As applications, we also obtained a large number of interesting results closely related to other results of Ramanujan. In the end, using Beta integral method, a large number of new and interesting hypergeometric identities are established. Known results earlier obtained by Choi et al. follow special cases of our main findings.

\[

_{2}F_{1}\left[

\begin{array}

[c]{ccc}%

a, & b; & \\

& & \frac{1+x}{2}\\

\frac{1}{2}(a+b+i+1); & &

\end{array}

\right]

\]

for $i=0,\pm1,\ldots,\pm9.$

For $i=0$, we have the well known, interesting and useful formula due to Kummer which was independently discovered by Ramanujan. The results are derived with the help of generalizations of Gauss's second summation theorem obtained recently by Rakha et al.

As applications, we also obtained a large number of interesting results closely related to other results of Ramanujan. In the end, using Beta integral method, a large number of new and interesting hypergeometric identities are established. Known results earlier obtained by Choi et al. follow special cases of our main findings.

This work is licensed under a Creative Commons Attribution 3.0 License.

Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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