A SUMMATION FORMULA FOR THE SERIES _{3}F_{2} DUE TO FOX AND ITS GENERALIZATIONS

- Journal title : Communications of the Korean Mathematical Society
- Volume 30, Issue 2, 2015, pp.103-108
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2015.30.2.103

Title & Authors

A SUMMATION FORMULA FOR THE SERIES _{3}F_{2} DUE TO FOX AND ITS GENERALIZATIONS

Choi, Junesang; Rathie, Arjun K.;

Choi, Junesang; Rathie, Arjun K.;

Abstract

Fox [2] presented an interesting identity for which is expressed in terms of a finite summation of `s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox [2] found a very interesting and general summation formula for as a special case of his above-mentioned general identity with the help of Kummer`s second summation theorem for . Here, in this paper, we show how two general summation formulas for , m being a nonnegative integer and i any integer, can be easily established by suitably specializing the above-mentioned Fox`s general identity with, here, the aid of generalizations of Kummer`s second summation theorem for obtained recently by Rakha and Rathie [7]. Several known results are also seen to be certain special cases of our main identities.

Keywords

Gamma function;Pochhammer symbol;hypergeometric function;generalized hypergeometric function;Kummer`s second summation theorem;Fox`s identity;

Language

English

References

1.

W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935; Reprinted by Stechert Hafner, New York, 1964.

2.

C. Fox, The expression of hypergeometric series in terms of a similar series, Proc. London Math. Soc. 26 (1927), 201-210.

3.

Y. S. Kim, M. A. Rakha, and A. K. Rathie, Extensions of classical summation theorems for the series $_2F_1$ , $_3F_2$ and $_4F_3$ with applications in Ramanujan's summations, Int. J. Math. Math. Sci. 2010 (2010), ID-309503, 26 pages.

4.

Y. S. Kim and A. K. Rathie, Applications of generalized Gauss's second summation theorem for the series $_2F_1$ , Math. Commun. 16 (2011), 481-489.

5.

E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.

6.

A. K. Rathie and T. Pogany, New summation formula for $_3F_2(1/2)$ and Kummer-type II transformation, Math. Commun. 13 (2008), no. 1, 63-66.

7.

M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series $_2F_1$ and $_3F_2$ with applications, Integral Transforms Spec. Funct. 22 (2011), no. 11, 823-840.

8.

L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London, and New York, 1966.

9.

H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.