Generalized Rodriguez-Villegas supercongruences involving $p$-adic Gamma functions

Ji-Cai Liu

Published 2016-11-23Version 1

Let $p\ge 5$ be a prime and $\langle a\rangle_p$ denote the least non-negative integer $r$ with $a\equiv r \pmod{p}$. We mainly prove that for $\langle a\rangle_p\equiv 0 \pmod{2}$ \begin{align*} \sum_{k=0}^{p-1}\frac{\left(\frac{1}{2}\right)_k(-a)_k(a+1)_k}{(1)_k^3}\equiv (-1)^{\frac{p+1}{2}}\Gamma_p\left(-\frac{a}{2}\right)^2\Gamma_p\left(\frac{a+1}{2}\right)^2 \pmod{p^2}, \end{align*} where $(x)_k=x(x+1)\cdots (x+k-1)$ and $\Gamma_p(\cdot)$ denotes the $p$-adic Gamma function. This partially extends four Rodriguez-Villegas supercongruences for truncated hypergeometric series ${}_3F_2$.