# Reduction formulae for the Lauricella functions in several variables

## Abstract

The main objective of this paper is to show how one can obtain several interesting reduction formulae for Lauricella functions from a multiple hypergeometric series identity established earlier by Jaimini et al. The results are derived with the help of generalized Kummer’s second summation formulas obtained earlier by Lavoi et al. Some special cases of our main result are explicitly demonstrated.

MSC:33C70, 33C065, 33C90, 33C05.

## 1 Introduction and results required

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

${\alpha }_{j}\in \mathbb{C}\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,p\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\beta }_{j}\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}\phantom{\rule{1em}{0ex}}\left({\mathbb{Z}}_{0}^{-}:=\mathbb{Z}\cup \left\{0\right\}=\left\{0,-1,-2,\dots \right\}\right),$

the generalized hypergeometric function${}_{p}F_{q}$ 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], pp.71-72])

(1.1)

where

$\omega :=\sum _{j=1}^{q}{\beta }_{j}-\sum _{j=1}^{p}{\alpha }_{j}\phantom{\rule{1em}{0ex}}\left({\alpha }_{j}\in \mathbb{C}\left(j=1,\dots ,p\right);{\beta }_{j}\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}\left(j=1,\dots ,q\right)\right)$
(1.2)

and ${\left(\lambda \right)}_{n}$ is the Pochhammer symbol defined (for $\lambda \in \mathbb{C}$), in terms of the familiar gamma function Γ, by

${\left(\lambda \right)}_{n}:=\frac{\mathrm{\Gamma }\left(\lambda +n\right)}{\mathrm{\Gamma }\left(\lambda \right)}=\left\{\begin{array}{cc}1\hfill & \left(n=0\right),\hfill \\ \lambda \left(\lambda +1\right)\cdots \left(\lambda +n-1\right)\hfill & \left(n\in \mathbb{N}\right).\hfill \end{array}$
(1.3)

The generalized Lauricella series in several variables is defined and represented in the following manner (see, for example, [[3], p.37]; see also [4]):

$\begin{array}{rcl}{F}_{C:{D}^{\prime };\dots ;{D}^{\left(n\right)}}^{A:{B}^{\prime };\dots ;{B}^{\left(n\right)}}\left(\begin{array}{c}{z}_{1}\\ ⋮\\ {z}_{n}\end{array}\right)& \equiv & {F}_{C:{D}^{\prime };\dots ;{D}^{\left(n\right)}}^{A:{B}^{\prime };\dots ;{B}^{\left(n\right)}}\left(\begin{array}{r}\left[\left(a\right):{\theta }^{\prime },\dots ,{\theta }^{\left(n\right)}\right]:\\ \left[\left(c\right):{\psi }^{\prime },\dots ,{\psi }^{\left(n\right)}\right]:\end{array}\\ \begin{array}{rrr}\left[\left({b}^{\prime }:{\varphi }^{\prime }\right)\right];& \dots ;& \left[\left({b}^{\left(n\right)}\right):{\varphi }^{\left(n\right)}\right];\\ \left[\left({d}^{\prime }:{\delta }^{\prime }\right)\right];& \dots ;& \left[\left({d}^{\left(n\right)}\right):{\delta }^{\left(n\right)}\right];\end{array}{z}_{1},\dots ,{z}_{n}\right)\\ :=& \sum _{{m}_{1},\dots ,{m}_{n}=0}^{\mathrm{\infty }}\mathrm{\Lambda }\left({m}_{1},\dots ,{m}_{n}\right)\frac{{z}_{1}^{{m}_{1}}}{{m}_{1}!}\cdots \frac{{z}_{n}^{{m}_{n}}}{{m}_{n}!},\end{array}$
(1.4)

where, for convenience,

$\begin{array}{c}\mathrm{\Lambda }\left({m}_{1},\dots ,{m}_{n}\right)\hfill \\ \phantom{\rule{1em}{0ex}}:=\frac{{\prod }_{j=1}^{A}{\left({a}_{j}\right)}_{{m}_{1}{\theta }_{j}^{\prime }+\cdots +{m}_{n}{\theta }_{j}^{\left(n\right)}}{\prod }_{j=1}^{{B}^{\prime }}{\left({b}_{j}^{\prime }\right)}_{{m}_{1}{\varphi }_{j}^{\prime }}\cdots {\prod }_{j=1}^{{B}^{\left(n\right)}}{\left({b}_{j}^{\left(n\right)}\right)}_{{m}_{n}{\varphi }_{j}^{\left(n\right)}}}{{\prod }_{j=1}^{C}{\left({c}_{j}\right)}_{{m}_{1}{\psi }_{j}^{\prime }+\cdots +{m}_{n}{\psi }_{j}^{\left(n\right)}}{\prod }_{j=1}^{{D}^{\prime }}{\left({d}_{j}^{\prime }\right)}_{{m}_{1}{\delta }_{j}^{\prime }}\cdots {\prod }_{j=1}^{{D}^{\left(n\right)}}{\left({d}_{j}^{\left(n\right)}\right)}_{{m}_{n}{\delta }_{j}^{\left(n\right)}}},\hfill \end{array}$
(1.5)

the coefficients

$\begin{array}{c}{\theta }_{j}^{\left(k\right)}\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,A\right),\phantom{\rule{2em}{0ex}}{\varphi }_{j}^{\left(k\right)}\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,{B}^{\left(k\right)}\right),\phantom{\rule{2em}{0ex}}{\psi }_{j}^{\left(k\right)}\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,C\right),\hfill \\ {\delta }_{j}^{\left(k\right)}\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,{D}^{\left(k\right)}\right)\phantom{\rule{1em}{0ex}}\left(\mathrm{\forall }k\in \left\{1,\dots ,n\right\}\right);\hfill \end{array}$

are real and nonnegative, and $\left(a\right)$ abbreviates the array of A parameters ${a}_{1},\dots ,{a}_{A}$; $\left({b}^{\left(k\right)}\right)$ abbreviates the array of ${B}^{\left(k\right)}$ parameters

${b}_{j}^{\left(k\right)}\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,{B}^{\left(k\right)};\mathrm{\forall }k\in \left\{1,\dots ,n\right\}\right),$

with similar interpretations for $\left({c}^{\left(k\right)}\right)$, etc.

In the course of study of hypergeometric functions of two or more variables, Srivastava [5, 6], Buschman and Srivastava [7], Grosjean and Sharma [8] and Grosjean and Srivastava [9] established a large number of double and multiple series identities involving essentially arbitrary coefficients (see, for example, [10]). Later Jaimini et al. [11] presented three substantially more general multiple series identities involving similar coefficients, one of which is recalled here as in the following theorem (see [[11], Theorem 3]).

Theorem 1Let$\mathrm{\Omega }\left(m\right)$represent a single-valued, bounded and real or complex function of the nonnegative integer-valued parameterm. Then we have

$\begin{array}{c}\sum _{{m}_{1},\dots ,{m}_{r}=0}^{\mathrm{\infty }}\mathrm{\Omega }\left({m}_{1}+\cdots +{m}_{r}\right)\frac{{\left(\alpha \right)}_{{m}_{1}+{m}_{2}}}{{\left(\alpha \right)}_{{m}_{1}}{\left(\alpha \right)}_{{m}_{2}}}\prod _{j=1}^{r}\left\{\frac{{\left({\mu }_{j}\right)}_{{m}_{j}}}{{m}_{j}!}{x}^{{m}_{j}}\right\}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{m,n=0}^{\mathrm{\infty }}\mathrm{\Omega }\left(m+2n\right){\left({\mu }_{1}+\cdots +{\mu }_{r}+2n\right)}_{m}\frac{{\left({\mu }_{1}\right)}_{n}{\left({\mu }_{2}\right)}_{n}}{{\left(\alpha \right)}_{n}}\frac{{x}^{m+2n}}{m!n!},\hfill \end{array}$
(1.6)

provided that each of the series involved is absolutely convergent.

From Theorem 1, with

$\mathrm{\Omega }\left(n\right)=\frac{{\prod }_{j=1}^{p}{\left({a}_{j}\right)}_{n}}{{\prod }_{j=1}^{q}{\left({b}_{j}\right)}_{n}}\phantom{\rule{1em}{0ex}}\left(n\in {\mathbb{N}}_{0}\right),$

we arrive at the following multiple hypergeometric identity involving the generalized Lauricella function defined by (1.4) (see [[11], Equation (3.1)]):

(1.7)

For $p=q=1$, (1.7) reduces at once to (see [[11], Equation (3.2)])

(1.8)

Finally, if we use Kummer’s second summation theorem (see, for example, [[12], p.11, Equation 2.4(2)]; see also [[13], Equation (1.4)])

${}_{2}F_{1}\left[\begin{array}{r}\alpha ,\beta ;\\ \frac{1}{2}\left(\alpha +\beta +1\right);\end{array}\frac{1}{2}\right]=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }\left[\frac{1}{2}\left(\alpha +\beta +1\right)\right]}{\mathrm{\Gamma }\left[\frac{1}{2}\left(\alpha +1\right)\right]\mathrm{\Gamma }\left[\frac{1}{2}\left(\beta +1\right)\right]}$
(1.9)

in (1.8) when

$x=\frac{1}{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}b=\frac{1}{2}\left(a+{\mu }_{1}+\cdots +{\mu }_{r}+1\right),$

Jaimini et al. [[11], Equation (3.6)] established the following interesting reduction formula for the generalized Lauricella function:

(1.10)

Here, in this paper, we aim mainly at showing how one can obtain several interesting reduction formulae for Lauricella functions from a multiple hypergeometric series identity (1.8). For this, we recall the following generalization of Kummer’s second summation theorem (1.9) obtained earlier by Lavoie et al. [14]:

$\begin{array}{c}{}_{2}F_{1}\left[\begin{array}{r}a,b;\\ \frac{1}{2}\left(a+b+\ell +1\right);\end{array}\frac{1}{2}\right]\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{a}{2}+\frac{b}{2}+\frac{\ell }{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{a}{2}-\frac{b}{2}-\frac{\ell }{2}+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{a}{2}-\frac{b}{2}+\frac{|\ell |}{2}+\frac{1}{2}\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\cdot \left(\frac{{A}_{\ell }}{\mathrm{\Gamma }\left(\frac{a}{2}+\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{b}{2}+\frac{\ell }{2}+\frac{1}{2}-\left[\frac{\ell +1}{2}\right]\right)}+\frac{{B}_{\ell }}{\mathrm{\Gamma }\left(\frac{a}{2}\right)\mathrm{\Gamma }\left(\frac{b}{2}+\frac{\ell }{2}-\left[\frac{\ell }{2}\right]\right)}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(\ell =0,±1,±2,±3,±4,±5\right),\hfill \end{array}$
(1.11)

where, and in what follows, $\left[x\right]$ denotes (as usual) the greatest integer less than or equal to x. The coefficients ${A}_{\ell }$ and ${B}_{\ell }$ are tabulated below.

It is remarked in passing that Equation (1.9) was incorrectly attributed to Gauss by Bailey [[12], p.11, Equation 2.4(2)] (see, for details, [[13], p.853]).

## 2 Main reduction formulae

The eleven reduction formulae in the form of a single result to be established are given in the following theorem.

Theorem 2The following reduction formula holds true:

(2.1)

where$\ell =0,±1,±2,±3,±4,±5$and, here, the coefficients${A}_{\ell }$and${B}_{\ell }$can be obtained in replacingaandbin Table 1by${\mu }_{1}+\cdots +{\mu }_{r}+2n$and$a+2n$, respectively.

Proof The proof is quite straightforward. In fact, if we set $x=\frac{1}{2}$ and $b=\frac{1}{2}\left(a+{\mu }_{1}+\cdots +{\mu }_{r}+\ell +1\right)$ in Equation (1.8), we have the following form:

(2.2)

Now, we observe that the ${}_{2}F_{1}$ appearing on the right-hand side of (2.2) can be evaluated with the help of generalized Kummer’s second summation theorem (1.11) in replacing a and b by ${\mu }_{1}+\cdots +{\mu }_{r}+2n$ and $a+2n$, respectively. And, after a little simplification, we easily arrive at the right-hand side of our main formula (2.1). The completes the proof of Theorem 2. □

## 3 Special cases

It is easy to see that the special case of (2.1) when $\ell =0$ leads to Equation (1.10) due to Jaimini et al. [11]. Here we consider two interesting special cases of our main formula (2.1). Setting $\ell =-1$ and $\ell =1$ in (2.1), we find Equations (3.1) and (3.2), respectively:

(3.1)

and

(3.2)

Clearly Equations (3.1) and (3.2) are closely related to Equation (1.10). The other special cases of (2.1) can also be obtained.

## References

1. Rainville ED: Special Functions. Macmillan Company, New York; 1960. Reprinted by Chelsea Publishing Company, Bronx, New York (1971)

2. Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.

3. Srivastava HM, Karlsson PW: Multiple Gaussian Hypergeometric Series. Wiley, New York; 1985.

4. Choi J, Rathie AK:Relations between Lauricella’s triple hypergeometric function ${F}_{A}^{\left(3\right)}\left(x,y,z\right)$ and Exton’s function ${X}_{8}$. Adv. Differ. Equ. 2013., 2013: Article ID 34 10.1186/1687-1847-2013-34

5. Srivastava HM:On the reducibility of Appell’s function ${F}_{4}$. Can. Math. Bull. 1973, 16: 295–298. 10.4153/CMB-1973-049-9

6. Srivastava HM: Some generalizations of Carlson’s identity. Boll. Unione Mat. Ital., A 1981, 18(5):138–143.

7. Buschman RG, Srivastava HM: Series identities and reducibility of Kampé de Fériet functions. Math. Proc. Camb. Philos. Soc. 1982, 91: 435–440. 10.1017/S0305004100059478

8. Grosjean CC, Sharma RK: Transformation formulae for hypergeometric series in two variables (II). Simon Stevin 1988, 62: 97–125.

9. Grosjean CC, Srivastava HM: Some transformation and reduction formulas for hypergeometric series in several variables. J. Comput. Appl. Math. 1991, 37: 287–299. 10.1016/0377-0427(91)90125-4

10. Srivastava HM, Raina RK: Some combinatorial series identities. Math. Proc. Camb. Philos. Soc. 1984, 96: 9–13. 10.1017/S0305004100061880

11. Jaimini BB, Koul CL, Srivastava HM: Some multiple series identities. Comput. Math. Appl. 1994, 28(4):19–24. 10.1016/0898-1221(94)00123-5

12. Bailey WN Cambridge Tracts in Mathematics and Mathematical Physics 32. In Generalized Hypergeometric Series. Cambridge University Press, Cambridge; 1935. Reprinted by Stechert-Hafner Service Agency, New York and London (1964)

13. Choi J, Rathie AK, Srivastava HM: A generalization of a formula due to Kummer. Integral Transforms Spec. Funct. 2011, 22: 851–859. 10.1080/10652469.2011.588786

14. Lavoie JL, Grondin F, Rathie AK:Generalizations of Whipple’s theorem on the sum of a ${}_{3}F_{2}$. J. Comput. Appl. Math. 1996, 72: 293–300. 10.1016/0377-0427(95)00279-0

## Acknowledgements

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 (2010-0011005).

## Author information

Authors

### Corresponding author

Correspondence to Junesang Choi.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors have equal contributions to each part of this paper. All authors have read and approved the final manuscript.

## Rights and permissions

Reprints and permissions

Choi, J., Rathie, A.K. Reduction formulae for the Lauricella functions in several variables. J Inequal Appl 2013, 312 (2013). https://doi.org/10.1186/1029-242X-2013-312