On Two Extension Formulas for Lauricella’s Function of the Second Kind of Several Variables

Ahmed Ali Atash¹, Ahmed Ali Al-Gonah²

¹Department of Mathematics, Faculty of Education-Shabwah, Aden University, Aden, Yemen.
²Department of Mathematics, Faculty of Education-Aden, Aden University, Aden, Yemen.

DOI: 10.4236/jamp.2016.43062                

Abstract

The aim of this research paper is to derive two extension formulas for Lauricella’s function of the second kind of several variables with the help of generalized Dixon’s theorem on the sum of the series  obtained by Lavoie et al. [1]. Some special cases of these formulas are also deduced.

Keywords

Extension Formulas, Lauricella’s Function, Dixon’s Theorem, Hypergeometric Functions

Received 1 February 2016; accepted 27 March 2016; published 30 March 2016

1. Introduction

The Lauricella’s function is defined and represented as follows [2]

(1.1)

;

where denotes the Pochhammer’s symbol defined by

(1.2)

(1.3)

Also, we note that

(1.4)

(1.5)

(1.6)

(1.7)

The generalized Lauricella’s function of several variables is defined as follows [2]

(1.8)

where

(1.9)

the coefficients for all

are real and positive; abbreviates the array of A parameters; abbreviate

the array of parameters for all with similar inter pretations for and . Note that, when the coefficients in Equation (1.8) equal to 1, the generalized Lauricella function (1.8) reduces to the following multivariable extension of the Kamp’e de F’eriet function [2] :

(1.10)

where

. (1.11)

In the theory of hypergeometric series, classical summation theorems such as Dixon, Watson and Whipple for the series, have many generalizations and wide applications; see for example [1] [3] - [6] . In the present investigation, we shall require the following generalization of the classical Dixon’s theorem for the series [1] :

(1.12)

,

where denotes the greatest integer less than or equal to x and denotes the usual absolute value of x. The coefficients are given respectively in [1] . When, (1.12) reduces immediately to the classical Dixon's theorem [3] , (see also [6] )

(1.13)

2. Extension Formulas

In this section, the following two extension formulas for Lauricella’s function of the second kind of several variables will be established:

(2.1)

and

(2.2)

where

(2.3)

(2.4)

(2.5)

for

The coefficients and can be obtained from the tables of and given in [1] by replacing a by and respectively.

Proof of (2.1): Denoting the left hand side of (2.1) by S, expanding in a power series and using the results [2] :

(2.6)

(2.7)

and, (2.8)

we get

(2.9)

where

(2.10)

Separating (2.9) into its even and odd terms, we have

(2.11)

Finally, in (2.11) if we use the result (1.12), then we obtain the right hand side of (2.1). This completes the proof of (2.1). The result (2.2) can be proved by the similar manner.

3. Special Cases

1) In (2.1), if we take and use the results (1.3)-(1.7), then after some simplification we obtain the following transformation formula:

(3.1)

which for, reduces immediately to a known result of Bailey [7]

(3.2)

where is Appell’s function [2] .

2) Similarly, in (2.2) if we take and use the results (1.3)-(1.7), then we obtain the following transformation formula:

(3.3)

3) In (2.2) if we take, then we get a known extension formulas [8] for Lauricella’s function of three va-

riables for.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Lavoie, J.L., Grondin, F., Rathie, A.K. and Arora, K. (1994) Generalizations of Dixon’s Theorem on the Sum of a . Mathematics of Computation, 62, 267-276.
[2]	Srivastava, H.M. and Manocha, H.L. (1984) A Treatise on Generating Functions. Hasted Press, New York.
[3]	Bailey, W.N. (1935) Generalized Hypergeometric Series. Cambridge University Press, Cambridge.
[4]	Lavoie, J.L., Grondin, F. and Rathie, A.K. (1992) Generalizations of Watson’s Theorem on the Sum of a . Indian Journal of Mathematics, 34, 23-32.
[5]	Lavoie, J.L., Grondin, F. and Rathie, A.K. (1996) Generalizations of Whipple’s Theorem on the Sum of a . Journal of Computational and Applied Mathematics, 72, 293-300. http://dx.doi.org/10.1016/0377-0427(95)00279-0
[6]	Rainville, E.D. (1960) Special Functions. The Macmillan Company, New York.
[7]	Bailey, W.N. (1953) On the Sum of Terminating . Quarterly Journal of Mathematics Oxford, 2, 237-240. http://dx.doi.org/10.1093/qmath/4.1.237
[8]	Atash, A.A. (2015) An Extension Formulas of Lauricella’s Functions by Applications of Dixon’s Summation Theorem. Applications and Applied Mathematics, 10, 1007-1018.