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A local limit theorem for generalized Stirling numbers. (English) Zbl 0727.60024

Authors’ summary: We consider triangular arrays \(S^ n_ k(a)\) of reals defined by the inversion \(x^ n=\sum^{n}_{k=0}S^ n_ k(a)(x-a_ 0)\cdots (x-a_ k),\) \(n=0,1,\dots\), and establish the local limit theorem in case \(a_ 0,a_ 1,\dots\) is an arithmetic progression. Additionally, the numbers \(\sum^{n}_{k=0}S^ n_ k(a)\) are asymptotically evaluated. For a geometric progression \(a_ 0,a_ 1,\dots\) not even a central limit theorem does hold.

MSC:

60F05 Central limit and other weak theorems
11B73 Bell and Stirling numbers
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