## 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