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An extension of Stirling numbers. (English) Zbl 0863.11012

The author discusses a definition of Stirling numbers \(s(n,k)\) and \(S(n,k)\) when \(n\) is a negative integer and \(k\) a positive one, through the formulas \((x)_n= \sum^\infty_{k=0} s(-n,k)x^k\) for \(n\geq 1\), and \(x^{-n}= \sum^\infty_{k=1} S(-n,k)(x)_k\) for \(n\geq 0\), where \((x)_n= \Gamma(x+1)/ \Gamma(x-n+1)\) is a rational function of \(x\) for \(n\leq-1\). These Stirling numbers are shown to occur in the computations of probabilities in some urn models which are variants of well known models for the classical Stirling numbers. Some properties of these generalized Stirling numbers are derived, such as recurrence formulas, generating functions, or relations like \[ \begin{aligned} S(-n,k) &= (-1)^{n+k-1} s(-k,n)\\ \text{or Stirling's formula} S(-n,k) &= {1\over k!}\sum^k_{r=1} (-1)^{k-r}{k\choose r}r^{-n}.\end{aligned} \] {}.
Reviewer: Ph.Biane (Paris)

MSC:

11B73 Bell and Stirling numbers
60C05 Combinatorial probability
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