Relations between Lauricella’s triple hypergeometric function $F_A^{(3)}(x,y,z)$ and Exton’s function $X_8$

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

$${\alpha }_j\in \mathbb{C}(j=1,\dots ,p)\text{and}{\beta }_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}({\mathbb{Z}}_0^{-}:={\mathbb{Z}}^{-}\cup \{0\}=\{0,-1,-2,\dots \}),$$

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]):

(1.1)

where

$$\omega :=\sum _{j=1}^q{\beta }_j-\sum _{j=1}^p{\alpha }_j({\alpha }_j\in \mathbb{C}(j=1,\dots ,p);{\beta }_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}(j=1,\dots ,q))$$

(1.2)

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

$${(1)}_n=n!(n\in {\mathbb{N}}_0),$$

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

$${(\lambda )}_{\nu }:=\frac{\mathrm{\Gamma }(\lambda +\nu )}{\mathrm{\Gamma }(\lambda )}=\{\begin{array}{cc}1\hfill & (\nu =0;\lambda \in \mathbb{C}\setminus \{0\}),\hfill \\ \lambda (\lambda +1)\cdots (\lambda +n-1)\hfill & (\nu =n\in \mathbb{N};\lambda \in \mathbb{C}),\hfill \end{array}$$

(1.3)

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

$$\begin{array}{rcl}{F}_A^{(3)}(x,y,z)& =& F_A^{(3)}(a,b_1,b_2,b_3;c_1,c_2,c_3;x,y,z)\\ :=& \sum _{m,n,p=0}^{\mathrm{\infty }}\frac{{(a)}_{m+n+p}{(b_1)}_m{(b_2)}_n{(b_3)}_p}{{(c_1)}_m{(c_2)}_n{(c_3)}_p}\frac{x^m}{m!}\frac{y^n}{n!}\frac{z^p}{p!}\\ (|x|+|y|+|z|<1;c_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}(j=1,2,3)).\end{array}$$

(1.4)

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

(1.5)

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]:

(2.1)

and

(2.2)

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.

Table 1 Contiguous relation coefficients

3 Main transformation formulae

The results to be established here are as follows:

(3.1)

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

$$S=\sum _{m,n,p=0}^{\mathrm{\infty }}\frac{{(a)}_{m+n+p}{(c_1)}_m{(b_1)}_n{(b_2)}_p}{{(2c_1+j)}_m{(c_2)}_n{(c_3)}_p}\frac{x^m}{m!}\frac{y^n}{n!}\frac{z^p}{p!}{2}^{2m}{(1+2x)}^{-(a+m+n+p)}.$$

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

$$S=\sum _{m,n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{m+n+p}{(c_1)}_m{(b_1)}_n{(b_2)}_p}{{(2c_1+j)}_m{(c_2)}_n{(c_3)}_p}\frac{x^{m+r}}{m!r!}\frac{y^n}{n!}\frac{z^p}{p!}{2}^{2m+r}{(-1)}^r{(a+m+n+p)}_r.$$

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

$$S=\sum _{m,n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{m+n+p+r}{(c_1)}_m{(b_1)}_n{(b_2)}_p}{{(2c_1+j)}_m{(c_2)}_n{(c_3)}_p}\frac{x^{m+r}}{m!r!}\frac{y^n}{n!}\frac{z^p}{p!}{2}^{2m+r}{(-1)}^r.$$

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

$$\sum _{r=0}^{\mathrm{\infty }}\sum _{m=0}^{\mathrm{\infty }}A(m,r)=\sum _{r=0}^{\mathrm{\infty }}\sum _{m=0}^{r}A(m,r-m),$$

after a little simplification, we have

$$S=\sum _{n,p,r=0}^{\mathrm{\infty }}\sum _{m=0}^r\frac{{(a)}_{n+p+r}{(c_1)}_m{(b_1)}_n{(b_2)}_p}{{(2c_1+j)}_m{(c_2)}_n{(c_3)}_p}\frac{x^r}{m!}\frac{y^n}{n!}\frac{z^p}{p!}\frac{2^{m+r}{(-1)}^{r-m}}{(r-m)!}.$$

Using the following formula:

$$(r-m)!=\frac{{(-1)}^{m}r!}{{(-r)}_m}(0\leqq m\leqq r;r,m\in {\mathbb{N}}_0),$$

after a little simplification, we get

$$S=\sum _{n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{n+p+r}{(b_1)}_n{(b_2)}_p{(-2)}^r}{{(c_2)}_n{(c_3)}_p}\frac{x^r}{r!}\frac{y^n}{n!}\frac{z^p}{p!}\sum _{m=0}^r\frac{{(c_1)}_m{(-r)}_m}{{(2c_1+j)}_{m}m!}{2}^m.$$

Using the definition of ${}_{p}F_q$ in (1.1) for the inner series, we obtain

$$S=\sum _{n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{n+p+r}{(b_1)}_n{(b_2)}_p{(-2)}^r}{{(c_2)}_n{(c_3)}_p}\frac{x^r}{r!}\frac{y^n}{n!}{\frac{z^p}{p!}}_2{F}_1\left[\begin{array}{c}-r,c_1;\\ 2c_1+j;\end{array}2\right].$$

Separating r into even and odd integers, we have

$$\begin{array}{rcl}S& =& \sum _{n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{n+p+2r}{(b_1)}_n{(b_2)}_p{2}^{2r}}{{(c_2)}_n{(c_3)}_p}\frac{x^{2r}}{(2r)!}\frac{y^n}{n!}{\frac{z^p}{p!}}_2{F}_1\left[\begin{array}{c}-2r,c_1;\\ 2c_1+j;\end{array}2\right]\\ +\sum _{n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{n+p+2r+1}{(b_1)}_n{(b_2)}_p{(-2)}^{2r+1}}{{(c_2)}_n{(c_3)}_p}\frac{x^{2r+1}}{(2r+1)!}\frac{y^n}{n!}{\frac{z^p}{p!}}_2{F}_1\left[\begin{array}{c}-2r-1,c_1;\\ 2c_1+j;\end{array}2\right].\end{array}$$

Making use of the following identity:

$${(\alpha )}_{2r}=2^{2r}{(\alpha )}_r{\left(\frac{\alpha }{2}\right)}_r,$$

after a little simplification, we get

$$\begin{array}{rcl}S& =& \sum _{n,p,r=0}^{\mathrm{\infty }}\frac{{(a)}_{n+p+2r}{(b_1)}_n{(b_2)}_p}{{(c_2)}_n{(c_3)}_p{(\frac{1}{2})}_r}\frac{x^{2r}}{r!}\frac{y^n}{n!}{\frac{z^p}{p!}}_2{F}_1\left[\begin{array}{c}-2r,c_1;\\ 2c_1+j;\end{array}2\right]\\ -2xa\sum _{n,p,r=0}^{\mathrm{\infty }}\frac{{(a+1)}_{n+p+2r}{(b_1)}_n{(b_2)}_p}{{(c_2)}_n{(c_3)}_p{(\frac{3}{2})}_r}\frac{x^{2r}}{r!}\frac{y^n}{n!}{\frac{z^p}{p!}}_2{F}_1\left[\begin{array}{c}-2r-1,c_1;\\ 2c_1+j;\end{array}2\right].\end{array}$$

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 $j=0$.

(4.1)

The case $j=1$.

(4.2)

The case $j=-1$.

(4.3)

The case $j=2$.

(4.4)

The case $j=-2$.

(4.5)

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.