Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers

Let Sᵣ(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."