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 $$\sum _{k=0}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{x}{k}\right)\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)(-4{)}^k}{(dk+1)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}(\mathrm{mod}{p}^2)$$ and $$\sum _{k=0}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{x}{k}\right)\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)(-2{)}^k}{(dk+1)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}(\mathrm{mod}{p}^2),$$ where $d\in \{0,1,2\}$. As consequences, we extend some known results. For example, for $p>3$ we show $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{3k}{k}\right){\left(\frac{4}{27}\right)}^k\equiv \frac{1}{9}+\frac{8}{9}p+\frac{4}{27}p{E}_{p-2}\left(\frac{1}{3}\right)(\mathrm{mod}{p}^2),$$ where $E_n(x)$ denotes the Euler polynomial of degree $n$. This generalizes a known congruence of Z.-W. Sun.

22 pages