Two congruences concerning Ap\'{e}ry numbers

TL;DR

This paper proves two conjectures by Z.-W. Sun concerning congruence properties of Apéry numbers, involving harmonic and Bernoulli numbers, for prime moduli.

Contribution

The paper confirms two conjectures about congruences of Apéry numbers related to harmonic and Bernoulli numbers for primes.

Findings

Confirmed Sun's conjecture for sum involving (2k+1)A_k modulo p^6.

Confirmed Sun's conjecture for sum involving (2k+1)^3A_k modulo p^9.

Established new congruence relations for Apéry numbers with prime moduli.

Abstract

Let $n$ be a nonnegative integer. The $n$-th Ap\'{e}ry number is defined by $$ A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $$ Z.-W. Sun ever investigated the congruence properties of Ap\'{e}ry numbers and posed some conjectures. For example, Sun conjectured that for any prime $p\geq7$ $$ \sum_{k=0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $$ and for any prime $p\geq5$ $$ \sum_{k=0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $$ where $H_n=\sum_{k=1}^n1/k$ denotes the $n$-th harmonic number and $B_0,B_1,\ldots$ are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.