Three supercongruences for Apery numbers or Franel numbers

Abstract

The Ap\'ery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=\sum_{k=0}^{n}{\binom{n+k}{2k}}^2{\binom{2k}{k}}^2\ \ \ \ \ {\rm and }\ \ \ \ \ \ f_n=\sum_{k=0}^{n}{\binom{n}{k}}^3(n=0, 1, \cdots,).$$ In this paper, we prove three supercongruences for Ap\'ery numbers or Franel numbers conjectured by Z.-W. Sun. Let $p\geq 5$ be a prime and let $n\in \mathbb{Z}^{+}$. We show that \begin{align} \notag \frac{1}{n}\bigg(\sum_{k=0}^{pn-1}(2k+1)A_k-p\sum_{k=0}^{n-1}(2k+1)A_k\bigg)\equiv0\pmod{p^{4+3\nu_p(n)}} \end{align} and \begin{align}\notag \frac{1}{n^3}\bigg(\sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3\sum_{k=0}^{n-1}(2k+1)^3A_k\bigg)\equiv0\pmod{p^{6+3\nu_p(n)}}, \end{align} where $\nu_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have \begin{align} \notag \frac{1}{n^3}\bigg(\sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2\sum_{k=0}^{n-1}(3k+2)(-1)^kf_k\bigg)\equiv0\pmod{p^{3}}. \end{align}