Explicit upper bounds for the Stirling numbers of the first kind

Abstract

We give explicit upper bounds for the Stirling numbers of the first kind s(n,m) which are asymptotically sharp. The form of such bounds varies according to m lying in the central or non-central regions of {1,…,n}. In each case, we use a different probabilistic representation of s(n,m) in terms of well known random variables to show the corresponding upper bounds. Some applications concerning the Riemann zeta function and a certain subset of the Comtet numbers of the first kind are also provided.