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Asymptotic expansions for the Stirling numbers of the first kind. (English) Zbl 0833.05005

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.

MSC:

05A16 Asymptotic enumeration
11B73 Bell and Stirling numbers
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References:

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