Supercongruences for sums involving $\binom ak^m$

TL;DR

This paper establishes new supercongruences involving sums of binomial coefficients raised to powers, using the WZ method, extending known results to higher powers and moduli.

Contribution

It introduces novel supercongruences for sums involving binomial coefficients and rational p-adic integers, proved via the Wilf-Zeilberger (WZ) method, extending previous congruence results.

Findings

Proved congruences for sums involving inom{a}{k}^2(-1)^k(1-rac{2a}{k}) modulo p^2.

Established congruences for inom{a}{k}^r(1-rac{2a}{k})^s modulo p^4 for specific r and s.

Extended supercongruence results to higher powers and moduli using the WZ method.

Abstract

Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0\pmod p$. In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \binom ak^2(-1)^k(1-\frac 2ak)$ modulo $p^2$ and $\sum_{k=0}^{p-1} \binom ak^r(1-\frac 2ak)^s$ modulo $p^4$, where $r\in\{3,4\}$ and $s\in\{1,3\}$.