Supercongruences for sums involving ${\left(\genfrac{}{}{0ex}{}{a}{k}\right)}^m$

Abstract

Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0(\mathrm{mod}p)$. In this paper, using WZ method we establish the congruences for ${\sum }_{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{a}{k}\right)}^2(-1{)}^k(1-\frac{2}{a}k)$ modulo $p^2$ and ${\sum }_{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{a}{k}\right)}^r(1-\frac{2}{a}k{)}^s$ modulo $p^4$, where $r\in \{3,4\}$ and $s\in \{1,3\}$.

34 pages