Combinatorial congruences and Stirling numbers

Abstract

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence $\sum_{j=0}^l{l\choose j}S(j,m)a^{l-j}(l=m,m+1,...)$ modulo a prime $p$, where $a$ and $m>0$ are integers, and those $S(j,m)$ are Stirling numbers of the second kind. We also give a new extension of Glaisher's congruence by showing that $(p-1)p^{[\log_p m]}$ is a period of the sequence $\sum_{j=r(mod p-1)}{l\choose j}S(j,m)(l=m,m+1,...)$ modulo $p$.