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# Title: Generalized Stirling Numbers I

Abstract: We consider generalized Stirling numbers of the second kind $% S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right)$, $% k=0,1,\ldots .rp+\sum_{s=2}^{L}r_{s}p_{s}$, where $a,b,\alpha_{s},\beta_{s}$ are complex numbers, and $r,p,r_{s},p_{s}$ are non-negative integers given, $s=2,\ldots ,L$. (The case $a=1,b=0,r=1,r_{s}p_{s}=0$, corresponds to the standard Stirling numbers $S\left( p,k\right)$.) The numbers $% S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right)$ are connected with a generalization of Eulerian numbers and polynomials we studied in previous works. This link allows us to propose (first, and then to prove, specially in the case $r=r_{s}=1$) several results involving our generalized Stirling numbers, including several families of new recurrences for Stirling numbers of the second kind. In a future work we consider the recurrence and the differential operator associated to the numbers $% S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right)$.
 Subjects: Combinatorics (math.CO); Number Theory (math.NT) MSC classes: 11B73 Cite as: arXiv:1803.05953 [math.CO] (or arXiv:1803.05953v1 [math.CO] for this version)

## Submission history

From: Claudio Pita Ruiz [view email]
[v1] Thu, 15 Mar 2018 19:07:33 GMT (15kb)