Congruences for the Apéry numbers modulo $p^3$

Abstract

Let $\{A_n^{\prime }\}$ be the Apéry numbers given by $A_n^{\prime }={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{n+k}{k}\right).$ For any prime $p\equiv 3(\mathrm{mod}4)$ we show that $A_{\frac{p-1}{2}}^{\prime }\equiv \frac{p^2}{3}{\left(\genfrac{}{}{0ex}{}{\frac{p-3}{2}}{\frac{p-3}{4}}\right)}^{-2}(\mathrm{mod}{p}^3)$. Let $\{t_n\}$ be given by $$t_0=1,t_1=5and{t}_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1{)}^2{t}_{n-1}(n\ge 1).$$ We also obtain the congruences for $t_p(\mathrm{mod}{p}^3),t_{p-1}(\mathrm{mod}{p}^2)$ and $t_{\frac{p-1}{2}}(\mathrm{mod}{p}^2)$, where $p$ is an odd prime.

16 pages