Certain new summation formulas for the series \(_4F_3 (1)\) with applications

Volume 9, Issue 6, pp 4722--4736 http://dx.doi.org/10.22436/jnsa.009.06.111

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Authors

Junesang Choi - Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea. Arjun K. Rathie - Department of Mathematics, School of Mathematical and Physical Sciences, Central University of Kerala, Riverside Transit Campus, Padennakkad P. O. Nileshwar, Kasaragod 671 328, Kerala, India.

Abstract

The main objective of this paper is to provide thirteen (presumably) new summation formulas for the series \(_4F_3\) of unit argument expressed in terms of Gamma functions. As special cases of our main results, we also present twenty four summation formulas for the terminating \(_4F_3 (1)\), whose further special cases are derived to give thirty two known summation formulas for the terminating \(_4F_3 (1)\). The results presented here are established with the help of a general result recorded in the book of Prudnikov et al. and the generalization of Watson's summation theorem obtained earlier by Lavoie et al.

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Keywords

• Gamma function
• Pochhammer symbol
• generalized hypergeometric function
• Watson's summation theorem
• generalized Watson's summation theorem.

MSC

•  33B15
•  33B20
•  33C20
•  33C05

References

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• [3] Y. S. Kim, M. A. Rakha, A. K. Rathie, Extensions of certain 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), 26 pages


• [4] J.-L. Lavoie, F. Grondin, A. K. Rathie, Generalizations of Watson's theorem on the sum of a \(_3F_2\), Indian J. Math., 34 (1992), 23--32


• [5] J.-L. Lavoie, F. Grondin, A. K. Rathie, Generalizations of Whipple's theorem on the sum of a \(_3F_2\), J. Comput. Appl. Math., 72 (1996), 293--300


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• [9] H. M. Srivastava, J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier, Amsterdam (2012)


• [10] H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Halsted Press, John Wiley and Sons, Inc., New York (1985)