## Abstract

The generalized Stirling numbers S_{s;h}(n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k; α,β, r) considered by Hsu and Shiue. From this relation, several properties of S_{s;h}(n, k) and the associated Bell numbers B_{s;h}(n) and Bell polynomials Bs_{;h{pipe}n}(x) are derived. The particular case s = 2 and h = -1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel polynomials is shown. The dual case s = -1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polyno- mials of negative order are introduced and studied. Finally, a q-analogue Ss;h(n, k{pipe}q) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the q-deformed numbers S_{s;h}(n, k{pipe}q) are special cases of the type-II p, q-analogue of generalized Stirling numbers introduced by Rem- mel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = -1 corresponding to the q-meromorphic Weyl algebra considered by Diaz and Pariguan.

Original language | English |
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Journal | Journal of Integer Sequences |

Volume | 15 |

Issue number | 8 |

State | Published - 2 Oct 2012 |

## Keywords

- Bell number
- Generalized Stirling number
- Generating function
- Touchard polynomial

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics