Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers

TL;DR

This paper proves the uniform convergence of an asymptotic expansion for r-associated Stirling numbers of the second kind across the entire parameter space, confirming a longstanding conjecture.

Contribution

It establishes the full uniform convergence of the asymptotic expansion for r-associated Stirling numbers, extending previous partial results.

Findings

Proved uniform convergence of the asymptotic expansion everywhere.

Extended previous partial convergence results to full convergence.

Confirmed a longstanding conjecture from Hennecart (1994).

Abstract

Let $S_r(p,q)$ be the $r$-associated Stirling numbers of the second kind, the number of ways to partition a set of size $p$ into $q$ subsets of size at least $r$. For $r=1$, these are the standard Stirling numbers of the second kind, and for $r=2$, these are also known as the Ward Numbers. This paper concerns asymptotic expansions of these Stirling numbers; such expansions have been known for many years. However, while uniform convergence of these expansions was conjectured in Hennecart's 1994 paper, it has not been fully proved. A recent paper (Connamacher and Dobrosotskaya, 2020) went a long way, by proving uniform convergence on a large set. In this paper we build on that paper and prove convergence "everywhere."