American Journal of Computational and Applied Mathematics

p-ISSN: 2165-8935    e-ISSN: 2165-8943

2013;  3(3): 182-185

doi:10.5923/j.ajcam.20130303.05

A Note on Hypergeometric Functions of One and Two Variables

Pankaj Srivastava, Priya Gupta

Department Name of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, 211004, India

Correspondence to: Priya Gupta, Department Name of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, 211004, India.

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Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.

Abstract

The present paper deals with transformations of hypergeometric function of one variable and Kampe de Feriet hypergeometric function of two variables by using transformations between two terminating Saalschutzian 4F3(1) series.

Keywords: Hypergeometric Functions, Transformation Formulae, Kampe de Feriet Hypergeometric Function

Cite this paper: Pankaj Srivastava, Priya Gupta, A Note on Hypergeometric Functions of One and Two Variables, American Journal of Computational and Applied Mathematics , Vol. 3 No. 3, 2013, pp. 182-185. doi: 10.5923/j.ajcam.20130303.05.

Article Outline

1. Introduction
2. Definitions and Notations
3. Main Result
4. Conclusions

1. Introduction

Hypergeometric function is a beautiful tool of special function that plays an important role in the field of analysis. The transformation theory played a major role to provide a platform for the development of beautiful transformation. It is important to mentation that whenever generalized hypergeometric function reduces to a gamma function, the results are very important from application point of view in mathematics, statistics and mathematical physics. On account of usefulness, hypergeometric function have already explored to the same extent by a number of special function experts notably C. F. G auss, M. M. Kummer, S. Ramanujan, B. C. Berndt[1], G. N. Watson[5], W. N. Bailey[18], E. D. Rainville[2], L. J. Slater[9], H. Exton[6], G. Gasper[4], M. Rahman, G. E. Andrews[3], R. P. Agrawl[13], R. Y. Denis et al.[15], S. N. Singh[16], H. M. Srivastava et al.[7], R. P. Singh[14], M. A. Rakha et al.[10], Yong S. Kim et al.[19], Pankaj Srivastava[11], R. K. Saxena et al.[12], S. P. Singh[17], Kung-Yu Chen et al.[8] etc. In the present paper transformations of hypergeometric function of one and two variables established this paper should be useful from the application point of view.

2. Definitions and Notations

A generalized hypergeometric function is defined as
Where
The series converges when none of denominator parameter are zero or negative integer. It is converges for all x when p<q, for |x|<1, it is converges when p=q+1and for |x|<1 it is converges when Re(∑bi-∑ai)>0 and it is only converges for x=0 when x=0 when p>q+1 unless it reduces to a polynomial.
A Kampe de Feriet hypergeometric function of two variables is defined as,
Where (ap) stands for the sequence of p parameters a1,…ap and |x|<1, |y|<1, l+m+n<p+q+k+1 for convergence.
We shall use the following known transformations and identity in our analysis as given in[1, 2, 4, 9]
(1)
([9], 2.5.11).
(2)
([9], 2.4.1.7).
(3)
([9], 2.5.22).
(4)
([9], 2.5.25).
(5)
([4], 3.1.1).
(6)
([1], Ch.10, eq. 6.3).
(7)
([2], Ch.4, lemma 10(2)).

3. Main Result

In this section we shall establish following results.
Theorem 3.1:
(8a)
(8b)
(8c)
Theorem 3.2:
(9a)
(9b)
(9c)
Theorem 3.3:
(10a)
(10b)
(10c)
Theorem 3.4:
(11a)
(11b)
(11c)
Theorem 3.5:
(12a)
(12b)
(12c)
Theorem 3.6:
(13a)
(13b)
(13c)
As an illustration we give proof of theorem (3.1).
Let Ωn is an arbitrary sequence of complex numbers, multiplying both sides of (1) by znΩn and summing over n from 0 to ∞ we get,
Making use of the identity (7) and (a-n)n=(-1)n(1-a)n in above we get after some simplification,
(14a)
Equation (14a) holds provided that both sides do exist. In particular, choosing and after some simplification we get,
(14b)
While choosing in eq. (14a) and after some simplification we get,
(14c)
Similarly one can establish theorems (3.2), (3.3), (3.4), (3.5) and (3.6) by suitable transformation from (2) to (6) and appropriate selection of Ωn.

4. Conclusions

The transformations of hypergeometric function of one and two variables established in this paper should be useful from the application point of view.

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