Supercongruences involving binomial coefficients and Euler polynomials

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.