Combinatorial congruences modulo prime powers

Let $p$ be any prime, and let $\alpha$ and $n$ be nonnegative integers. Let $r\in \mathbb{Z}$ and $f(x)\in \mathbb{Z}[x]$. We establish the congruence

$p^{\deg f}\sum _{k\equiv r\, (\operatorname{mod}p^{\alpha })} \bi... ...atorname{mod} p^{\sum _{i=\alpha }^{\infty } \lfloor n/{p^{i}}\rfloor }\right )$

(motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If $\alpha$ is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then

We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a $p$-adic order bound given by the authors in a previous paper can be attained when $p=2$.