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 $$\sum_{k=r(mod p^β)}\binom{n}{k}(-1)^k f([(k-r)/p^α]),$$ where $α\geβ\ge 0$, $n\ge p^{α-1}$ and $r\in Z$. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as $$\sum_{k=r(mod p^α)}\binom{n}{k} a^k\equiv 0 (mod p^{[(n-p^{α-1})/ϕ(p^α)]})$$ provided that $α>1$ and $a\equiv-1(mod p)$.