Asymptotic Normality of the $(r,\beta )$-Stirling Numbers

Abstract

In this paper, the unimodality of $(r,\beta )$-Stirling numbers and certain asymptotic approximation of $(r,\beta )$-Bell numbers are established. Together with these results and the most general form of Central Limit Theorem, viz. Bounded Variance Normal Convergence Criterion, the $(r,\beta )$-Stirling numbers are shown to be asymptotically normal.