Congruences for the Ap\'ery numbers modulo $p^3$

Abstract

Let $\{A'_n\}$ be the Ap\'ery numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be given by $$t_0=1,\ t_1=5\quad\hbox{and}\quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1).$$ We also obtain the congruences for $t_p\pmod {p^3},\ t_{p-1}\pmod {p^2}$ and $t_{\frac{p-1}2}\pmod {p^2}$, where $p$ is an odd prime.