Power Reducibility and Congruences

Rong-Hua Wang, Michael X. X. Zhong

Published 2023-01-05Version 1

In this paper, a criterion on the power reducibility of holonomic sequences is presented. As applications, we show Ap\'ery numbers $A_k$ and the central Delannoy polynomials $D_k(z)$ are both power reducible and present series of congruences. For example, when $p>3$ is a prime, we find that for each $r\in\mathbb{N}$, there is a $p$-adic integer $c_r$ such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2r+1}A_k\equiv c_r p \pmod {p^3}. \end{equation*}