Combinatorial congruences modulo prime powers

Abstract

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{°f}\sum _{k=r(mod{p}^a)}\left(\genfrac{}{}{0ex}{}{n}{k}\right)(-1{)}^{k}f((k-r)/p^a)=0(mod{p}^{{\sum }_{i=a}^{\infty }[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then $$\frac{1}{[n/p^{a-1}]!}\sum _{k=r(mod{p}^a)}\left(\genfrac{}{}{0ex}{}{pn+s}{pk+t}\right)(-1{)}^{pk}((k-r)/p^{a-1}{)}^l=\frac{1}{[n/p^{a-1}]!}\sum _{k=r(mod{p}^a)}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{s}{t}\right)(-1{)}^k((k-r)/p^{a-1}{)}^l(modp).$$ 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.