The p-adic analysis of Stirling numbers via higher order Bernoulli numbers

Abstract

In this paper, we use our previous study of the higher order Bernoulli numbers [Formula: see text] to investigate [Formula: see text]-adic properties of Stirling numbers of the second kind [Formula: see text]. For example we give a new greatly simplified proof of the formula [Formula: see text] if [Formula: see text], and generalize this result to arbitrary primes [Formula: see text]. We also extend these results to a translation invariance property. Finally, we consider the Stirling numbers of the first kind [Formula: see text], with new results analogous to those for the Stirling numbers of the second kind. New mod [Formula: see text] congruences for Stirling numbers of both kinds are also given.