On certain hypergeometric identities deducible by using the beta integral method
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RESEARCH
Open Access
On certain hypergeometric identities deducible by using the beta integral method Adel K Ibrahim1,3 , Medhat A Rakha2,3* and Arjun K Rathie4 * Correspondence: [email protected] 2 Department of Mathematics and Statistics - College of Science, Sultan Qaboos University, Alkhoud, P.O. Box 36, Muscat, 123, Oman 3 Permanent address: Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt Full list of author information is available at the end of the article
Abstract 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. MSC: Primary 33C05; 33C20; secondary 33C70 Keywords: hypergeometric series; Kummer summation theorem; beta integral
1 Introduction and preliminaries The generalized hypergeometric series p Fq is defined by [, ] ⎡ ⎤ α , . . . , αp ; ∞ n ⎢ ⎥ (α )n · · · (αp )n z = , F z ⎦ p q⎣ (β )n · · · (βq )n n! n= β , . . . , βq ;
()
where (a)n is the Pochhammer symbol (or the shifted or raised factorial, since ()n = n!) defined (for a ∈ C) by
(a)n =
=
⎧ ⎨,
n = ,
⎩a(a + ) · · · (a + n – ), n ∈ N = {, , . . .} (a + n) , (a)
a ∈ C \ Z–
()
and Z– denotes the set of non-positive integers, C the set of complex numbers, and (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 , . . . , αp and the denominator parameters β , . . . , βq take on complex values, provided that no zeros appear in the denominator of (), that is, βj ∈ C \ Z– ;
j = , . . . , q.
()
©2013 Ibrahim et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Ibrahim et al. Advances in Difference Equations 2013, 2013:341 http://www.advancesindifferenceequations.com/content/2013/1/341
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For the detailed conditions of the convergence of series (), we refer to []. It is not out of place to mention here that if one of the numerator parameters, say aj , is a negative integer, then series () reduces to a polynomial in z of degree –aj . It is interesting to mention here that whenever a generalized hypergeometric funct
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