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Extension of a quadratic transformation due to Whipple with an application
Advances in Difference Equations volume 2013, Article number: 157 (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.
The generalized hypergeometric function , with p numerator and q denominator parameters is defined by 
where denotes the Pochhammer symbol (or the shifted factorial, since ) defined for any complex number a by
Using the fundamental relation , can be written in the form
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].
In this research paper, we are interested in the following classical Saalschütz summation theorem :
By utilizing (1.4), Bailey  obtained the following three interesting quadratic transformations:
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).
Very recently, Rakha and Rathie  established the extension of Saalschütz summation theorem (1.4) in the form
Also, in the same paper , as an applications, they have obtained the following results:
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
The beta function is defined by the first integral and is known to be evaluated as the second one as follows:
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.
Thus, for example, let us consider (1.7) in the form
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).
Corollary 1 For a to be a negative integer, the following hypergeometric transformation holds true.
Following the same procedure, from (2.1) and known results (1.5) and (1.6), we get the following identities (presumably new).
Corollary 2 For a to be a negative integer, the following hypergeometric transformation holds true.
Corollary 3 For a to be a negative integer, the following hypergeometric transformation holds true.
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.
Theorem 1 The following extension of Whipple’s transformation (1.7) holds true.
where A is given by
Proof In order to establish (3.1), we proceed as follows. Denote the left-hand side of (3.1) by S, then upon expressing the as a series given by the definition (1.1), after some simplification, we have
Using the well-known binomial theorem
Now, replacing m by , making use of the result 
and then using the identity
we have, after some simplification,
Summing up the inner series, we have
Finally, using the extension of Saalschütz’s summation theorem (1.8) and after much simplification, we get
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. □
Remark If we equate the coefficients of on both sides of equation (3.1), we get the following interesting identity (presumably new):
where A is the same as given in (3.2).
Further, in (3.3), if we take , we get the following identity (presumably new):
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.
Corollary 4 For a to be a negative integer, the following hypergeometric transformation holds true.
where, of course, A is the same as defined in (3.2).
Here we mention two interesting special cases of (4.1).
In (4.1), if we replace a by and take and , we get the following transformation.
Corollary 5 For a to be a negative integer, the following hypergeometric transformation holds true.
In (4.1), if we take and , we get the following result.
Corollary 6 For a to be a negative integer, the following hypergeometric transformation holds true.
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.
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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.
The authors declare that they have no competing interests.
All authors contributed equally to this paper. They read and approved the final paper.
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Wang, X., Rathie, A.K. Extension of a quadratic transformation due to Whipple with an application. Adv Differ Equ 2013, 157 (2013). https://doi.org/10.1186/1687-1847-2013-157
- Whipple’s transformation
- Saalschütz’s summation theorem
- extension of a quadratic transformation
- beta integral