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Some theorems on generalized Stirling numbers. (English) Zbl 1101.11312
The author investigates the generalized Stirling numbers of the second kind $S(n,k;\alpha ,\beta ,\gamma )$ introduced by {\it L. C. Hsu} and {\it P. J.-S. Shiue} [Adv. Appl. Math. 20, 366-384 (1998; Zbl 0913.05006)]. He derives for them horizontal and vertical recurrence relations, a formula for the ordinary generating function $$\sum _{n\ge k}S(n,k;0,\beta ,\gamma )t^n=t^k/\prod _{j=0}^k (1-(\beta j+\gamma )t)$$ (here ${}_{j=0}$ should be probably replaced by ${}_{j=1}$) and explicit formulae. In the end the evaluation $$\sum _{n\ge k}S(n-1,k-1;\alpha ,\beta ,\gamma )/(x\mid \alpha )_n= 1/(x-\gamma \mid \beta )_k,$$ where $(x\mid \alpha )_n=x(x-\alpha )\ldots (x-(n-1)\alpha )$, is proved.

11B73Bell and Stirling numbers
05A10Combinatorial functions