Three supercongruences for Apery numbers or Franel numbers

Abstract

The Apéry numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n+k}{2k}\right)}^2{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\mathrm{and}{f}_n=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^3(n=0,1,\cdots ,).$$ In this paper, we prove three supercongruences for Apéry numbers or Franel numbers conjectured by Z.-W. Sun. Let $p\ge 5$ 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+3ν_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+3ν_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}

References in corpus (1)

• Supercongruences for sporadic sequences