Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach

Abstract

Using the saddle point method, we obtain from the generating function of the Stirling numbers of the first kind [n j] and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Goncharov theorem with more precision. In the region j = n - n(alpha); alpha > 1/2, we analyze the dependence of [n j] on alpha.

Keywords

• stirling numbers
• saddle point method
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]