Completely effective error bounds for Stirling Numbers of the first and second kind via Poisson Approximation
| Autor: | Stephen DeSalvo, Richard Arratia |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Stirling numbers of the first kind
010102 general mathematics Stein's method Stirling numbers of the second kind 06 humanities and the arts 0603 philosophy ethics and religion Poisson distribution 01 natural sciences Interpretation (model theory) Combinatorics symbols.namesake 060302 philosophy symbols FOS: Mathematics Discrete Mathematics and Combinatorics Stirling number Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics 05A16 05A20 60C05 11B73 Asymptotic expansion Mathematics |
| Popis: | We provide completely effective error estimates for Stirling numbers of the first and second kind, denoted by $s(n,m)$ and $S(n,m)$, respectively. These bounds are useful for values of $m \geq n - O(\sqrt{n})$. An application of our Theorem 5 yields, for example, \[ s(10^{12},\ 10^{12}-2\times 10^6)/10^{35664464} \in [ 1.87669, 1.876982 ], \] \[ S(10^{12},\ 10^{12}-2\times 10^6)/10^{35664463} \in [ 1.30121, 1.306975 ]. \] The bounds are obtained via Chen-Stein Poisson approximation, using an interpretation of Stirling numbers as the number of ways of placing non-attacking rooks on a chess board. As a corollary to Theorem 5, summarized in Proposition 1, we obtain two simple and explicit asymptotic formulas, one for each of $s(n,m)$ and $S(n,m)$, for the parametrization $m = n - t\, n^a$, $0 \leq a \leq \frac{1}{2}.$ These asymptotic formulas agree with the ones originally observed by Moser and Wyman in the range $0 Comment: 19 pages, 4 Figures, 19 References |
| Databáze: | OpenAIRE |
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