Uniform convergence of an asymptotic approximation to associated Stirling numbers

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 by Hennecart, it has not been fully proved. A recent paper by Connamacher and Dobrosotskaya went a long way by proving uniform convergence on a large set. In this paper, we build on that paper and prove convergence "everywhere"