On certain hypergeometric identities deducible by using the beta integral method
© Ibrahim et al.; licensee Springer. 2013
Received: 12 August 2013
Accepted: 29 October 2013
Published: 25 November 2013
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: 33C05, 33C20, 33C70.
1 Introduction and preliminaries
For the detailed conditions of the convergence of series (1), we refer to . It is not out of place to mention here that if one of the numerator parameters, say , is a negative integer, then series (1) reduces to a polynomial in z of degree .
It is interesting to mention here that whenever a generalized hypergeometric function reduces to a 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 , Watson, Dixon, Whipple and Saalschütz for the series 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 , 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 in the product must be expressible in terms of gamma functions.
This result was independently rediscovered by Ramanujan [, Entry 2, p.49].
provided 2a or d is a nonpositive integer.
The coefficients and
4(b + 3 + 2n)
b + 2 + 4n
−(3b + 6 + 4n)
−(b + 1 + 2n)
1 − b − 2n
1 − b − 4n
3 − 3b − 4n
4(1 − b − 2n)
Here, in this paper, we show how one can easily obtain eleven interesting hypergeometric identities including the Krattenthaler-Rao result (5) in the form of a single unified result by employing the beta integral method developed by Krattenthaler and Rao . The results are derived with the help of generalization (6) of Kummer’s formula (4). Several interesting special cases of our main result including (5) are also explicitly demonstrated.
The results presented in this paper are simple, interesting, easily established and (potentially) useful.
2 Main result
Our eleven main identities are given here in the form of a single unified result asserted in the following theorem.
for . The coefficients and are the same as given in Table 1.
Finally, equating (8) and (9), we get the desired result (7). This completes the proof of (7). The above restriction on the parameters d and e may now be removed by appeal to analytic continuation. □
3 Special cases
Here we shall consider some of the very interesting special cases of our main result (7). Each of the following formulas holds true provided 2a or d must be a non-positive integer.
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (10). □
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (11). □
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (12). □
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (13). □
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (14). □
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (15). □
Proof Setting in (7) and simplifying the resulting identity, we are led to formula (16). □
Result (10) is the well-known result of Krattenthaler and Rao  and results (11) to (16) are closely related to it.
Remark 9 We conclude this section by mentioning that the series appearing on the right-hand side of results (13) to (16) can also be written as a sum of two , and then we can obtain alternate forms of the results.
The authors are thankful to the referee for making certain very useful suggestions. The work of this research paper was supported by the research grant (05/4/33) funded by Jazan University - Jazan, Saudi Arabia.
- Bailey WN: Generalized Hypergeometric Series. Cambridge University Press, Cambridge; 1964.MATHGoogle Scholar
- Rainville ED: Special Functions. Chelsea Publishing, New York; 1971.MATHGoogle Scholar
- Bailey WN: Products of generalized hypergeometric series. Proc. Lond. Math. Soc. 1928, 28: 242-254.View ArticleMathSciNetMATHGoogle Scholar
- Kummer EE:Uber die hypergeometrische reihe . J. Reine Angew. Math. 1836, 15: 39-83.MathSciNetView ArticleGoogle Scholar
- Berndt BC: Ramanujan’s Notebooks, Part II. Springer, Berlin; 1989.View ArticleMATHGoogle Scholar
- Krattenthaler C, Rao KS: Automatic generation of hypergeometric identities by the beta integral method. J. Comput. Appl. Math. 2003, 160: 159-173. 10.1016/S0377-0427(03)00629-0MathSciNetView ArticleMATHGoogle Scholar
- Kim YS, Rakha MA, Rathie AK: Generalizations of Kummer’s second theorem with applications. Comput. Math. Math. Phys. 2010, 50: 387-402. 10.1134/S0965542510030024MathSciNetView ArticleMATHGoogle Scholar
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