Power-Partible Reduction and Congruences

Abstract

Based on the polynomial reduction, a holonomic (or, P-recursive) sequence $F(k)$ can be decomposed into a summable part and a reduced part. In this paper, we show that when $F(k)$ has a certain kind of symmetry, the reduced part contains only odd or even powers. The reduction in this case is called a power-partible reduction, which is then applied to obtain new series of congruences for Ap\'ery numbers $A_k$ and central Delannoy polynomials $D_k(z)$. In particular, when $p>3$ is a prime, we prove that for each r\in\bN, 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*