On congruences involving Apéry numbers

Abstract

In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime $p>3$ and positive odd integer $m$, we prove that there is a $p$-adic integer $c_m$ only depending on $m$ such that $$\sum _{k=0}^{p-1}(2k+1{)}^m(-1{)}^k{A}_k\equiv c_{m}p\left(\frac{p}{3}\right)(\mathrm{mod}{p}^3),$$ where $A_k={\sum }_{j=0}^k{\left(\genfrac{}{}{0ex}{}{k}{j}\right)}^2{\left(\genfrac{}{}{0ex}{}{k+j}{j}\right)}^2$ is the Apéry number and $(\frac{\cdot }{p})$ is the Legendre symbol.

12 pages