On certain hypergeometric identities deducible by using beta integral method

The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was used successfully and systematically by Krattenthaler and Rao in their well known, very interesting research papers. The results are derived with the help of generalization of a quadratic transformation formula due to Kummer very recently obtained by Kim, et al. . Several identities including one obtained earlier by Krattenthaler and Rao follow special cases of our main findings. The results established in this paper are simple, interesting, easily established and may be potentially useful.


Introduction and Preliminaries
The generalized hypergeometric series p F q is defined by [2,7]  and Z − 0 denotes the set of nonpositive integers, C the set of complex numbers, and Γ (a) is the familiar Gamma function. Here p and q are positive integers or zero (interpreting an empty product as unity), and we assume for simplicity that the variable z, the numerator parameters α 1 , . . . , α p and the denominator parameters β 1 , . . . , β q take on complex values, provided that no zeros appear in the denominator of (1.1), that is For the detailed conditions of the convergence of the series (1.1), we refer to [7]. It is not out of place to mention here that if one of the numerator parameters, say a j is a negative integer, then the series (1.1) reduces to a polynomial in z of degree −a j .
It is interesting to mention here that whenever a generalized hypergeometric functions reduces to gamma function, the results are very important from the application point of view. Thus the classical summation theorems such as those of Gauss, Gauss second, Kummer and Bailey for the series 2 F 1 ; Watson, Dixon, Whipple and Saalschütz for the series 3 F 2 and others play an important role in the theory of hypergeometric and generalized hypergeometric series.
In a very interesting, popular and useful research paper, Bailey [1], by employing the above mentioned classical summation theorems, obtained a large number of results involving products of generalized hypergeometric series as well as quadratic and cubic transformations. Several other results were also given by Gauss and Kummer.
Evidently, if the product of two generalized hypergeometric series can be expressed as another generalized hypergeometric series with argument x, the coefficients of x n in the product must be expressible in terms of gamma functions.
In our present investigation, we are interested in the following quadratic transformation due to Kummer [6] ( Very recently, Kim, et al. [4] have obtained the following generalization of the Kummer quadratic transformation formula (1.4) in the form Here, as usual, [x] denotes the greatest integer less than or equal to x and its modulus is denoted by |x|. The coefficients, A j and B j are given in the following table.
Here, in this paper, we show how one can easily obtain eleven interesting hypergeometric identities including the Krattenthaler-Rao result (1.5) in the form of a single unified result by employing the beta integral method developed by Krattenthaler and Rao [5]. The results are derived with the help of the generalization (1.6) of the Kummer's formula (1.4). Several interesting special cases of our main result including (1.5) are also explicitly demonstarted.
The results presented in this paper are simple, interesting, easily established and (potentially) useful.

Main Result
Our eleven main identities are given here in the form of a single unified result asserted in the following theorem.   Proof. Let us first assume that a be a non-positive integer. Multiply the left-hand side of (1.6) by x d−1 (1 − x) e−d−1 and integrating the resulting equation with respect to x from 0 to 1, expressing the involved 2 F 1 as series and changing the order of integration and summation, which is easily seen to be justified due to the uniform convergence of the involved series, we have Evaluating the beta-integral and using the identity we have, after some algebra summing up the series, we have (2.2) Now, multiply the right-hand side of (1.6) by x d−1 (1 − x) e−d−1 and integrating with respect to x from 0 to 1, we have after some simplification Evaluating the beta integrals and using the Legendre's duplication formula we have after some simplification Finally, equating (2.3) and (2.3), we get the desired result (2.1). This completes the proof of (2.1).

Special Cases
Here we shall consider some of the very interesting special cases of our main result (2.1). Each of the following formulas hold true provided a or d must be a nonpositive integer.
Proof. Setting j = 0 in (2.1) and simplifying the resulting identity, we are led to the formula (3.1).
Proof. Setting j = 1 in (2.1) and simplifying the resulting identity, we are led to the formula (3.2).
Proof. Setting j = −1 in (2.1) and simplifying the resulting identity, we are led to the formula (3.3). Proof. Setting j = 2 in (2.1) and simplifying the resulting identity, we are led to the formula (3.4).