Supercongruences involving binomial coefficients and Euler polynomials

TL;DR

This paper proves new supercongruences involving binomial coefficients and Euler polynomials modulo prime squares, extending previous results and providing explicit formulas for sums related to binomial coefficients and Euler polynomials.

Contribution

It establishes novel supercongruences for binomial coefficient sums involving Euler polynomials, generalizing known congruences and extending the scope of supercongruence results.

Findings

Proves supercongruences modulo p^2 for specific binomial sums.

Extends known results to new cases involving Euler polynomials.

Provides explicit congruences for sums of binomial coefficients with rational functions.

Abstract

Let $p$ be an odd prime and let $x$ be a $p$-adic integer. In this paper, we establish supercongruences for $$ \sum_{k=0}^{p-1}\frac{\binom{x}{k}\binom{x+k}{k}(-4)^k}{(dk+1)\binom{2k}{k}}\pmod{p^2} $$ and $$ \sum_{k=0}^{p-1}\frac{\binom{x}{k}\binom{x+k}{k}(-2)^k}{(dk+1)\binom{2k}{k}}\pmod{p^2}, $$ where $d\in\{0,1,2\}$. As consequences, we extend some known results. For example, for $p>3$ we show $$ \sum_{k=0}^{p-1}\binom{3k}{k}\left(\frac{4}{27}\right)^k\equiv\frac19+\frac89p+\frac{4}{27}pE_{p-2}\left(\frac13\right)\pmod{p^2}, $$ where $E_n(x)$ denotes the Euler polynomial of degree $n$. This generalizes a known congruence of Z.-W. Sun.