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Relations between Lauricella’s triple hypergeometric function and Exton’s function
Advances in Difference Equations volume 2013, Article number: 34 (2013)
Very recently Choi et al. derived some interesting relations between Lauricella’s triple hypergeometric function and the Srivastava function by simply splitting Lauricella’s triple hypergeometric function into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function and Exton’s function in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating series which were very recently obtained by Kim et al. and also include the relationship between and due to Exton.
MSC: 33C20, 44A45.
1 Introduction and preliminaries
In the usual notation, let ℂ denote the set of complex numbers. For
and is the Pochhammer symbol or the shifted factorial since
which is defined (for ), in terms of the familiar gamma function Γ, by
it being understood conventionally that .
By using the notations as described in [, p.33, Equation 1.4(1)] (with ) (see also [, p.114, Equation (1)], [, Equation (2.1)] and [, p.60, Equation 1.7(1)]), Lauricella’s triple hypergeometric function is defined by
Motivated essentially by the works by Lardner  and Carlson , by simply splitting Lauricella’s triple hypergeometric function into eight parts, very recently Choi et al.  presented several relationships between and the Srivastava function (see also ). The widely-investigated Srivastava’s triple hypergeometric function , which was introduced over four decades ago by Srivastava [, p.428] (see also , [, p.44, Equation 1.5(14)] and [, p.69, Equation 1.7(39)]), provides an interesting unification (and generalization) of Lauricella’s 14 triple hypergeometric functions (see , [, pp.113-114]) and Srivastava’s three functions , and (see [, pp.99-100]; see also [14–16], [, p.43] and [, pp.60-68]).
In fact, in 1982 Exton  published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as and and introduced a set of 20 distinct triple hypergeometric functions to and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions , , a Humbert function and a Humbert function in their kernels. It is not out of place to mention here that Exton’s functions to have been studied a lot until today; see, for example, the works [12, 18–23] and . Moreover, Exton  presented a large number of very interesting transformation formulas and reducible cases with the help of two known results which are called in the literature Kummer’s first and second transformations or theorems.
Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function and Exton’s function in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating series which were very recently obtained by Kim et al.  and also include the relationship between and due to Exton .
3 Main transformation formulae
The results to be established here are as follows:
where the coefficients and can be obtained by simply changing n and α into r and , respectively, in Table 1 of and .
Proof For convenience and simplicity, by denoting the left-hand side of (3.1) by S and using the series definition of as given in (1.4), after a little simplification, we have
Using the binomial theorem (see, for example, [, p.58]) for the last factor, we get
Using the identity , after a little simplification, we obtain
after a little simplification, we have
Using the following formula:
after a little simplification, we get
Using the definition of in (1.1) for the inner series, we obtain
Separating r into even and odd integers, we have
Making use of the following identity:
after a little simplification, we get
Finally, using the known results (2.1) and (2.2), after a little simplification, we easily arrive at the right-hand side of (3.1). This completes the proof of (3.1). □
4 Special cases
In our main formula (3.1), if we take and ±2, after a little simplification, and interpret the respective resulting right-hand sides with the definition of Exton’s triple hypergeometric series given in (1.5), we get the following very interesting relations between and :
The case .
The case .
The case .
The case .
The case .
Remark Clearly, Equation (4.1) is Exton’s result (see ) and Equations (4.2) to (4.5) are closely related to it. The other special cases of (3.1) can also be expressed in terms of in a similar manner.
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Dedicated to Professor Hari M Srivastava.
This paper was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012-0002957).
The authors declare that they have no competing interests.
The authors have equal contributions to each part of this paper. All authors have read and approved the final manuscript.
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Choi, J., Rathie, A.K. Relations between Lauricella’s triple hypergeometric function and Exton’s function . Adv Differ Equ 2013, 34 (2013). https://doi.org/10.1186/1687-1847-2013-34
- gamma function
- hypergeometric functions of several variables
- multiple Gaussian hypergeometric series
- Exton’s triple hypergeometric series
- Gauss’s hypergeometric functions
- Lauricella’s triple hypergeometric functions