Combinatorial congruences modulo prime powers

Abstract:

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 \begin{equation*}p^{\deg f}\sum _{k\equiv r (\operatorname {mod}p^{\alpha })} \binom nk(-1)^{k}f\left (\frac {k-r}{p^{\alpha }}\right ) \equiv 0\ \left (\operatorname {mod} p^{\sum _{i=\alpha }^{\infty } \lfloor n/{p^{i}}\rfloor }\right )\end{equation*} (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 \begin{equation*}\begin {split} &\frac {1}{\lfloor n/p^{\alpha -1}\rfloor !} \sum _{k\equiv r (\operatorname {mod}p^{\alpha })}\binom {pn+s}{pk+t}(-1)^{pk} \left (\frac {k-r}{p^{\alpha -1}}\right )^{l}\\ \equiv & \frac {1}{\lfloor n/p^{\alpha -1}\rfloor !} \sum _{k\equiv r (\operatorname {mod} p^{\alpha })}\binom nk\binom st(-1)^{k} \left (\frac {k-r}{p^{\alpha -1}}\right )^{l} \ (\operatorname {mod} p). \end{split}\end{equation*} 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 $p=2$.