A summation formula for the series ${}_3F_2$ due to Fox and its generalizations
Commun. Korean Math. Soc. 2015 Vol. 30, No. 2, 103-108
https://doi.org/10.4134/CKMS.2015.30.2.103
Printed April 30, 2015
Junesang Choi and Arjun K. Rathie
Dongguk University, Central University of Kerala, Riverside Transit Campus
Abstract : Fox \cite{Fox} presented an interesting identity for ${}_pF_q$ which is expressed in terms of a finite summation of ${}_pF_q$'s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox \cite{Fox} found a very interesting and general summation formula for ${}_3F_2(1/2)$ as a special case of his above-mentioned general identity with the help of Kummer's second summation theorem for ${}_2F_1(1/2)$. Here, in this paper, we show how two general summation formulas for $$ {}_{3}F_{2} \left[ \aligned \alpha,\, \beta,\,\gamma &;\\ \alpha-m,\, \frac{1}{2}(\beta+\gamma+i+1) &; \endaligned \,\, \frac{1}{2} \right],$$ $m$ 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 ${}_2F_1(1/2)$ obtained recently by Rakha and Rathie \cite{Ra-Ra}. 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
MSC numbers : Primary 33B20, 33C20; Secondary 33B15, 33C05
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