Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials

TL;DR

This paper proves several conjectures by Z.-W. Sun regarding congruences involving Apery polynomials, specifically focusing on sums with alternating signs and their divisibility properties for integer values of x.

Contribution

The paper provides rigorous proofs for multiple conjectures of Z.-W. Sun related to congruences of sums involving Apery polynomials, advancing understanding in this area.

Findings

Confirmed conjectures on congruences for Apery polynomial sums

Established divisibility properties for sums with alternating signs

Extended knowledge on congruences for polynomial sequences

Abstract

The Apery polynomials are defined by $A_n(x)=\sum_{k=0}^{n}{n\choose k}^2{n+k\choose k}^2 x^k$ for all nonnegative integers $n$. We confirm several conjectures of Z.-W. Sun on the congruences for the sum $\sum_{k=0}^{n-1}(-1)^k(2k+1) A_k(x)$ with $x\in Z$.