Extension of a quadratic transformation due to Whipple with an application
© Wang and Rathie; licensee Springer 2013
Received: 7 March 2013
Accepted: 20 May 2013
Published: 4 June 2013
The aim of this research is to provide an extension of an interesting and useful quadratic transformation due to Whipple. The result is derived with the help of extension of classical Saalschütz’s summation theorem recently added in the literature. The transformation is further used to obtain a new hypergeometric identity by employing the so-called beta integral method introduced and studied systematically by Krattenthaler and Rao.
MSC:33C20, 33C05, 33B20.
KeywordsWhipple’s transformation Saalschütz’s summation theorem extension of a quadratic transformation beta integral
where Γ is the well-known gamma function.
It is well known that whenever a generalized hypergeometric function reduces to quotient of the products of the gamma function, the results are very important from the application point of view. Thus in the theory of hypergeometric and generalized hypergeometric series, summation formulas and transformation formulas play an important role.
In a very popular, interesting and useful research article, Bailey , by employing classical summation theorems such as those of Gauss, Gauss second, Kummer and Bailey for the series ; Watson, Dixon, Whipple and Saalschütz for the series , established a large number of very interesting results (known as well as new) involving products of generalized hypergeometric series.
It is not out of place to mention here that recently a good deal of progress has been done in the direction of generalizing the above mentioned classical summation theorems. For details, we refer to [3–5].
The transformation formula (1.7) is originally due to Whipple  who obtained it by other means.
It is interesting to mention here that in (1.7), (i) if we replace a by and take and and (ii) if we take and and simplify, we respectively recover (1.5) and (1.6).
The results (1.9) and (1.11) may be regarded as the extensions of (1.5) and (1.6) as it can be seen by taking .
The aim of this research is twofold. First, by utilizing the extension of Saalschütz’s summation theorem (1.8), we obtain a natural extension of Whipple’s transformation (1.7). Then, by employing the beta integral method, we obtain a new hypergeometric identity. The results derived in this paper are simple, easily established and may be potentially useful.
2 Demonstration of the beta integral method
Krattenthaler and Rao  made a systematic use of the so-called beta integral method, a method of deriving new hypergeometric identities from old ones by mainly using the beta integral in (2.1) based on the Mathematica Package HYP, to illustrate several interesting identities for the hypergeometric series and Kampé de Fériet series in most cases of unit arguments.
In this section, we also apply the beta integral method to the known results (1.7), (1.9) and (1.11) to get new hypergeometric identities. However, we shall derive one identity in detail and others can be obtained similarly.
Now, multiplying both sides of equation (2.2) by , integrating the resulting equation with respect to x from 0 to 1, expressing the involved as series, changing the order of integration and summation (which is easily seen to be justified due to the uniform convergence of the series involved in the process) and using the beta integral (2.1), then after some simplification, summing up the resulting series, we get the following identity (presumably new).
Following the same procedure, from (2.1) and known results (1.5) and (1.6), we get the following identities (presumably new).
We conclude this section by remarking that the results (2.4) and (2.5) can also be obtained from (2.3) by (i) replacing a by and taking and and (ii) taking and respectively.
3 Extension of Whipple’s transformation (1.7)
The extension of Whipple’s quadratic transformation (1.7) to be established in this paper is given in the following theorem.
where A is the same as given in (3.2). Finally, summing up the series with the definition (1.1), we easily arrive at the right-hand side of (3.1). This completes the proof of Theorem 1. □
where A is the same as given in (3.2).
which can also be obtained directly by equating coefficients of in Whipple’s transformation (1.7).
We remark in passing that if in Theorem 1 we take , so that , after little simplification, we recover Whipple’s transformation (1.7).
As already explained in detail in Section 2, using the beta integral method to our main result (3.1), it is not difficult to obtain the following hypergeometric identity which is given here without a proof.
where, of course, A is the same as defined in (3.2).
In (4.1), if we replace a by and take and , we get the following transformation.
In (4.1), if we take and , we get the following result.
We conclude this section by remarking that the results (4.2) and (4.3) can also be obtained from (1.9) and (1.11) by applying the beta integral method.
This work was, in part, supported by the National Natural Science Foundation of China (No. 11201291), Natural Science Foundation of Shanghai (No. 12ZR1443800) and a grant of ‘The First-class Discipline of Universities in Shanghai’. The authors are grateful to the worthy referees for giving certain very useful suggestions.
- Rainville ED: Special Functions. Macmillan, New York; 1960. Reprinted by Chelsea Publishing, Bronx, New York (1971)Google Scholar
- Bailey WN: Products of generalized hypergeometric series. Proc. Lond. Math. Soc. 1928, 28(2):242-254.View ArticleGoogle Scholar
- Kim YS, Rakha MA, Rathie AK:Extensions of certain classical summation theorems for the series , and with applications in Ramanujan’s summations. Int. J. Math. Math. Sci. 2010., 2010: Article ID 3095031Google Scholar
- Rakha MA, Rathie AK:Generalizations of classical summation theorems for the series and . Integral Transforms Spec. Funct. 2011, 22(11):823-840. 10.1080/10652469.2010.549487MathSciNetView ArticleGoogle Scholar
- Vidnunas R: A generalization of Kummer identity. Rocky Mt. J. Math. 2002, 32(2):919-936. 10.1216/rmjm/1030539701View ArticleGoogle Scholar
- Whipple FJW: Some transformations of generalized hypergeometric series. Proc. Lond. Math. Soc. 1927, 26(2):257-272.MathSciNetView ArticleGoogle Scholar
- Rakha MA, Rathie AK: Extensions of Euler type II transformation and Saalschütz’s theorem. Bull. Korean Math. Soc. 2011, 48(1):151-156.MathSciNetView ArticleGoogle 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 ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.