Asymptotics of the Stirling numbers of the second kind revisited

Abstract

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Stirling numbers of the second kind {n / m} and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Gaussian theorem with more precision. In the region m = n - na, 1 > a > 1/2, we analyze the dependence of {n / m} on a. An extension of some Moser and Wyman's result to full m range is also provided. This paper fits within the framework of Analytic Combinatorics.