Hwang, Hsien-Kuei Asymptotic expansions for the Stirling numbers of the first kind. (English) Zbl 0833.05005 J. Comb. Theory, Ser. A 71, No. 2, 343-351 (1995). Let \(s(n, m)\) denote the unsigned Stirling numbers of the first kind. For any \(\eta> 0\) and natural number \(v\), the following asymptotic formula holds uniformly \[ {s(n, m)\over n!}= {1\over n} \sum_{0\leq k\leq v} {\Pi_{m,k} (\log n)\over n^k}+ O \Biggl({(\log n)^m\over m!n^{v+ 2}}\Biggr) \] for \(1\leq m\leq \eta\log n\), where \(\Pi_{m, k}(x)\) are explicitly given polynomials in \(x\) of degree \(m- 1\). Since the asymptotic behaviour of \(\Pi_{m, k}(\log n)\) is unclear for \(m= \Omega(\log n)\), a uniform asymptotic expansion for \(\Pi_{m, 0}(\log n)\) is also given. These results can be interpreted as saying that the Stirling numbers of the first kind are asymptotically Poisson distributed of parameter \(\log n\). The proof uses a variant of the saddle point method. Reviewer: L.A.Székely (Budapest) Cited in 24 Documents MSC: 05A16 Asymptotic enumeration 11B73 Bell and Stirling numbers Keywords:Stirling numbers of the first kind; polynomials; uniform asymptotic expansion; saddle point method PDFBibTeX XMLCite \textit{H.-K. Hwang}, J. Comb. Theory, Ser. A 71, No. 2, 343--351 (1995; Zbl 0833.05005) Full Text: DOI References: [1] Comtet, L., Advanced Combinatorics, The Art of Finite and Infinite Expansions (1974), Reidel: Reidel Dordrecht, rev. and enlarged ed. [2] Evgrafov, M. A., Asymptotic Estimates and Entire Functions (1961), Gordon and Breach: Gordon and Breach New York, (A. L. Shields, Transl.) · Zbl 0121.30202 [3] Flajolet, P.; Odlyzko, A. M., Singularity analysis of generating functions, SIAM J. Discrete Math., 3, 216-240 (1990) · Zbl 0712.05004 [4] Flajolet, P.; Soria, M., Gaussian limiting distributions for the number of components in combinatorial structures, J. Combin. Theory Ser. A, 53, 165-182 (1990) · Zbl 0691.60035 [5] H.-K. Hwang; H.-K. Hwang [6] Luke, Y. L., (The Special Functions and Their Approximations, Vol. I (1969), Academic Press: Academic Press New York) [7] Selberg, A., Note on a paper by L. G. Sathe, J. Indian Math. Soc., 18, 83-87 (1954) · Zbl 0057.28502 [8] Temme, N. M., Asymptotic estimates of Stirling numbers, Stud. Appl. Math., 89, 233-243 (1993) · Zbl 0784.11007 [9] Tenenbaum, G., Introduction à la Théorie Analytique et Probabiliste des Nombres (1990), Institut Elie Cartan, Université de Nancy I · Zbl 0788.11001 [10] Wilf, H. S., The asymptotic behavior of the Stirling numbers of the first kind, J. Combin. Theory Ser. A, 64, 344-349 (1993) · Zbl 0795.05005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.