On divisibility of sums of Apery polynomials

TL;DR

This paper proves a divisibility property of sums involving powers of Apery polynomials, extending understanding of their algebraic structure and congruence relations.

Contribution

It establishes a new divisibility result for sums of Apery polynomials raised to a power, generalizing previous known congruences.

Findings

Sum of Apery polynomials with alternating signs divisible by n

Generalization of divisibility for polynomial sums with binomial coefficients

New congruence relations for Apery polynomial sums

Abstract

For any positive integers $m$ and $\alpha$, we prove that $$\sum_{k=0}^{n-1}\epsilon^k(2k+1)A_k^{(\alpha)}(x)^m\equiv0\pmod{n}, $$ where $\epsilon\in\{1,-1\}$ and $$ A_n^{(\alpha)}(x)=\sum_{k=0}^n\binom{n}{k}^{\alpha}\binom{n+k}{k}^{\alpha}x^k.$$