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

Medhat A Rakha,Arjun K Rathie,Adel K Ibrahim

doi:10.5539/jmr.v6n3p62

Medhat A Rakha, Arjun K Rathie + Show 1 more

Open Access

https://doi.org/10.5539/jmr.v6n3p62

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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.

Highlights

= 1, and which is of fundamental importance in the it is represented by the symbol 2F1[a, b; c; theory of x] or 2F1
A natural generalization of 2F1 is the generalized hypergeometric function, the so-called pFq, which is defined in the following manner (Rainville, 1971)
We conclude this section with the remain that the results presented in this paper have been verified numerically through the computer algebra system MATHEMATICA, a general system of doing mathematics by computer

Summary

Introduction

We are interested in mentioning the following Gauss’s second summation theorem (Bailey, 1964) viz. Applying Euler’s transformation formula (4) in the right-hand side of (9), we get the following another result due to Ramanujan (Berndt, 1989) viz. Remark 1 For Ramanujan results (9) and (10) see a note by Rakha et al (2013). In 1996, Lavoie et al (1996) have generalized the Gauss’s second summation theorem (3) and obtained explicit expressions of (14). ±9, and obtained nineteen results closely related to Gauss’s second summation Theorem (3) in the form of a single result which is given here.

Special Cases

Results

New Hypergeometric Identities