Two congruences concerning Apéry numbers

Abstract

Let $n$ be a nonnegative integer. The $n$-th Apéry number is defined by $$A_n:=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)}^2{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2.$$ Z.-W. Sun ever investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime $p\ge 7$ $$\sum _{k=0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}{p}^2{H}_{p-1}(\mathrm{mod}{p}^6)$$ and for any prime $p\ge 5$ $$\sum _{k=0}^{p-1}(2k+1{)}^3{A}_k\equiv p^3+4p^4{H}_{p-1}+\frac{6}{5}{p}^8{B}_{p-5}(\mathrm{mod}{p}^9),$$ where $H_n={\sum }_{k=1}^{n}1/k$ denotes the $n$-th harmonic number and $B_0,B_1,\dots$ are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.

14 pages