Polynomial extension of Fleck's congruence

Abstract

Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\sum _{k=r(mod{p}^{\beta })}\left(\genfrac{}{}{0ex}{}{n}{k}\right)(-1{)}^{k}f([(k-r)/p^{\alpha }]),$$ where $\alpha \ge \beta \ge 0$, $n\ge p^{\alpha -1}$ and $r\in Z$. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as $$\sum _{k=r(mod{p}^{\alpha })}\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^k\equiv 0(mod{p}^{[(n-p^{\alpha -1})/\varphi (p^{\alpha })]})$$ provided that $\alpha >1$ and $a\equiv -1(modp)$.

11 pages