On the asymptotic normality of the Legendre-Stirling numbers of the second kind

TL;DR

This paper establishes asymptotic normality and local limit theorems for Legendre-Stirling numbers of the second kind, extending previous results on Chebyshev-Stirling numbers and analyzing their unimodality.

Contribution

It provides new asymptotic formulas and normality results for Legendre-Stirling numbers, supplementing existing analyses of related Stirling numbers.

Findings

Asymptotic formulas derived for Legendre-Stirling numbers

Proven asymptotic normality and unimodality of modified Legendre-Stirling numbers

Extended analysis to include Chebyshev-Stirling numbers

Abstract

For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.