Asymptotics of Stirling and Chebyshev‐Stirling Numbers of the Second Kind

Abstract

For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev–Stirling numbers, a special case of the Jacobi–Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.

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

“…The proof for (1.5), which relies on a series of special properties of the Chebyshev-Stirling numbers does not work for the Jacobi-Stirling numbers in general (see also the remarks following Theorem 2.9 below). However, by a more elaborate analysis than in [15] it turns out that asymptotics being similar to (1.5) can be derived for the Legendre-Stirling numbers as well. More precisely, the main result of this paper is given by…”

“…Next, for our modified Eisenstein series we derive a useful error estimate (c.f. Lemma 5.1 in [15]). Lemma 2.5.…”

“…The main objects of our investigations are the Legendre-Stirling numbers, a special case of the Jacobi-Stirling numbers both of the second kind. Following the recent literature, e.g., [3], [9], [15], we denote the latter numbers by the curly bracket symbol n j γ , n, j being non-negative integers, and γ is a fixed non-negative parameter. A formal definition of these numbers can be given through the triangular recurrence relation [3], [15]…”

“…The Legendre differential operator corresponds to the case α = β = 0, i.e., γ = 1, and hence the numbers n j 1 are called Legendre-Stirling numbers of the second kind. Besides the already mentioned original field of differential equations during the past decade the Jacobi-Stirling numbers received considerable attention especially in combinatorics and graph theory, see, e.g., [1], [2], [3], [6], [7], [9], [15], [16], [17], [21], [22], [23]. Among the n j γ 's the Legendre-Stirling numbers n j 1…”

“…For the unique solution of the recurrence (1.1) the following explicit formula is known [3], [12], [15], [16]…”