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# Relations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$

*Advances in Difference Equations*
**volume 2013**, Article number: 34 (2013)

## Abstract

Very recently Choi *et al.* derived some interesting relations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and the Srivastava function ${F}^{(3)}[x,y,z]$ by simply splitting Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ into eight parts. Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$ in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating ${}_{2}F_{1}(2)$ series which were very recently obtained by Kim *et al.* and also include the relationship between ${F}_{A}^{(3)}(x,y,z)$ and ${X}_{8}$ due to Exton.

**MSC:** 33C20, 44A45.

## 1 Introduction and preliminaries

In the usual notation, let ℂ denote the set of *complex* numbers. For

the *generalized hypergeometric series* ${}_{p}F_{q}$ with *p* numerator parameters ${\alpha}_{1},\dots ,{\alpha}_{p}$ and *q* denominator parameters ${\beta}_{1},\dots ,{\beta}_{q}$ is defined by (see, for example, [[1], Chapter 4]; see also [[2], p.73]):

where

and ${(\lambda )}_{\nu}$ is the Pochhammer symbol or the *shifted factorial* since

which is defined (for $\lambda ,\nu \in \mathbb{C}$), in terms of the familiar gamma function Γ, by

it being *understood conventionally* that ${(0)}_{0}:=1$.

By using the notations as described in [[3], p.33, Equation 1.4(1)] (with $n=3$) (see also [[4], p.114, Equation (1)], [[5], Equation (2.1)] and [[6], p.60, Equation 1.7(1)]), Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ is defined by

Motivated essentially by the works by Lardner [7] and Carlson [8], by simply splitting Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ into eight parts, very recently Choi *et al.* [5] presented several relationships between ${F}_{A}^{(3)}(x,y,z)$ and the Srivastava function ${F}^{(3)}[x,y,z]$ (see also [9]). The widely-investigated Srivastava’s triple hypergeometric function ${F}^{(3)}[x,y,z]$, which was introduced over four decades ago by Srivastava [[10], p.428] (see also [5], [[3], p.44, Equation 1.5(14)] and [[6], p.69, Equation 1.7(39)]), provides an interesting unification (and generalization) of Lauricella’s 14 triple hypergeometric functions ${F}_{1},\dots ,{F}_{14}$ (see [11], [[12], pp.113-114]) and Srivastava’s three functions ${H}_{A}$, ${H}_{B}$ and ${H}_{C}$ (see [[13], pp.99-100]; see also [14–16], [[3], p.43] and [[6], pp.60-68]).

Exton’s triple hypergeometric function ${X}_{8}$ (see [17]; see also [[3], p.84, Entry 41a and p.101, List 41a]) is defined by

In fact, in 1982 Exton [17] 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 ${(a)}_{2m+2n+p}$ and ${(a)}_{2m+n+p}$ and introduced a set of 20 distinct triple hypergeometric functions ${X}_{1}$ to ${X}_{20}$ and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions ${}_{0}F_{1}$, ${}_{1}F_{1}$, a Humbert function ${\psi}_{1}$ and a Humbert function ${\varphi}_{2}$ in their kernels. It is not out of place to mention here that Exton’s functions ${X}_{1}$ to ${X}_{20}$ have been studied a lot until today; see, for example, the works [12, 18–23] and [24]. Moreover, Exton [17] 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 ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$ in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating ${}_{2}F_{1}(2)$ series which were very recently obtained by Kim *et al.* [25] and also include the relationship between ${F}_{A}^{(3)}(x,y,z)$ and ${X}_{8}$ due to Exton [17].

## 2 Results required

For our purpose, we require the following results recently obtained by Kim *et al.* [25]:

and

where $n\in {\mathbb{N}}_{0}$, $j=0$, $\pm 1,\dots ,\pm 5$, $[x]$ is the greatest integer less than or equal to *x* and its modulus is denoted by $|x|$, and the coefficients ${\mathcal{A}}_{j}$ and ${\mathcal{B}}_{j}$ are given in Table 1.

## 3 Main transformation formulae

The results to be established here are as follows:

where the coefficients ${\mathcal{C}}_{j}$ and ${\mathcal{D}}_{j}$ can be obtained by simply changing *n* and *α* into *r* and ${c}_{1}$, respectively, in Table 1 of ${\mathcal{A}}_{j}$ and ${\mathcal{B}}_{j}$.

*Proof* For convenience and simplicity, by denoting the left-hand side of (3.1) by *S* and using the series definition of ${F}_{A}^{(3)}(x,y,z)$ as given in (1.4), after a little simplification, we have

Using the binomial theorem (see, for example, [[2], p.58]) for the last factor, we get

Using the identity ${(a)}_{m+n+p}{(a+m+n+p)}_{r}={(a)}_{m+n+p+r}$, after a little simplification, we obtain

Now using the following well-known formal manipulation of double series (see [[2], p.56]; for other manipulations, see also [[26], Eq. (1.4)]):

after a little simplification, we have

Using the following formula:

after a little simplification, we get

Using the definition of ${}_{p}F_{q}$ 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 $j=0,\pm 1$ and ±2, after a little simplification, and interpret the respective resulting right-hand sides with the definition of Exton’s triple hypergeometric series ${X}_{8}$ given in (1.5), we get the following very interesting relations between ${F}_{A}^{(3)}(x,y,z)$ and ${X}_{8}$:

The case $j=0$.

The case $j=1$.

The case $j=-1$.

The case $j=2$.

The case $j=-2$.

**Remark** Clearly, Equation (4.1) is Exton’s result (see [17]) 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 ${X}_{8}$ in a similar manner.

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## Acknowledgements

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).

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Choi, J., Rathie, A.K. Relations between Lauricella’s triple hypergeometric function ${F}_{A}^{(3)}(x,y,z)$ and Exton’s function ${X}_{8}$.
*Adv Differ Equ* **2013, **34 (2013). https://doi.org/10.1186/1687-1847-2013-34

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### Keywords

- 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