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A unified approach to generalized Stirling numbers. (English) Zbl 0913.05006
Set $(z| \alpha)_n =z(z-\alpha) \cdots (z-n \alpha+ \alpha)$ and $(z | \alpha)_0=1$. For complex parameters $(\alpha, \beta,r) \ne(0,0,0)$, the paper studies the inverse relations $$(t|\alpha)_n=\sum^n_{k=0} S^1(n,k) (t-r | \beta)_k \quad \text{and} \quad (t | \beta)_n =\sum^n_{k=0} S^2(n,k) (t+r | \alpha)_k.$$ The resulting generalized Stirling numbers include binomial coefficients, Lah numbers, signless Stirling numbers, and many other generalized Stirling numbers. Generating functions and Dobinski-type formulae are also given.

05A15Exact enumeration problems, generating functions
11B73Bell and Stirling numbers
Full Text: DOI
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