A contiguous extension of Dixon's theorem for a terminating ${}_4F_3(1)$ series with applications

Authors: MOHAMMAD IDRIS QURESHI, RICHARD BRUCE PARIS, SHAKIR HUSSAIN MALIK

Abstract: We derive a summation formula for the terminating hypergeometric series \[{}_4F_3\left[\!\!\begin{array}{c}-m,a,b,1+c\\1+a+m,1+a-b,c\end{array}\!\!;1\right],\] where $m$ denotes a nonnegative integer. Using this summation formula, we establish a reduction formula for the Srivastava-Daoust double hypergeometric function with arguments $z$ and $-z$. Special cases of this reduction formula lead to several reduction formulas for the hypergeometric functions ${}_{p+1}F_p$ with quadratic arguments when $p=2,3$ and 4 by employing series rearrangement techniques. A general double series identity involving a bounded sequence of arbitrary complex numbers is also given.

Keywords: Hypergeometric summation theorems, Srivastava-Daoust double hypergeometric function, bounded sequence, series rearrangement technique

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