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The multiparameter noncentral Stirling numbers. (English) Zbl 0802.05003

Let \(\alpha= (\alpha_ 0,\alpha_ 1,\alpha_ 2,\dots)\) be a sequence of real numbers. The author defines Stirling numbers \(S(n,k,\alpha)\) and \(s(n,k,\alpha)\) satisfying the recurrences \[ S(n+1,k,\alpha)= S(n,k-1, \alpha)+ (k- \alpha_ n) S(n,k,\alpha) \] and \[ s(n+1,k,\alpha)= s(n,k- 1,\alpha)+ (\alpha_ k- n) s(n,k,\alpha) \] and derives some interesting properties. For \(\alpha_ k=0\) one gets the usual Stirling numbers, for \(\alpha_ k= \alpha\) the noncentral Stirling numbers of M. Koutras [Discrete Math. 42, 73-89 (1982; Zbl 0506.10009)] and for \(\alpha_ k= q^ k-1\) a \(q\)-analog of Stirling numbers considered by the reviewer [Sitzungsber. Abt. II, Ă–sterr. Akad. Wiss. Math.-Naturwiss. Kl. 201, No. 1-10, 97-109 (1992)].
Reviewer: J.Cigler (Wien)

MSC:

05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
11B73 Bell and Stirling numbers
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

Zbl 0506.10009
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