Supercongruences for a truncated hypergeometric series

Abstract

The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum _{k=1}^{p-1}\frac{(x{)}_k(1-x{)}_k}{(1{)}_k^2}\cdot \frac{1}{k^a}$$ for $a=1$ and $a=2$. In the last section, the special case $x=1/2$ is considered.

Revised version