Power-Partible Reduction and Congruences for Apéry Numbers

Abstract

In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Apéry numbers $A_k$. In particular, we prove that, for any $r\in \mathbb{N}$, there exists an integer ${\tilde{c}}_r$ such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2r+1}A_k\equiv \tilde{c}_r p \pmod {p^3} \end{equation*} holds for any prime $p>3$.