Communications of the
Korean Mathematical Society CKMS

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