Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866911828093698048 |
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| author | Komatsu, Takao Bagno, Eli Garber, David |
| author_facet | Komatsu, Takao Bagno, Eli Garber, David |
| contents | The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion-exclusion principle, and also exponential generating functions.
These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized using results of Comtet and Lancaster.
We generalize these theorems, which present equivalent conditions for a triangle of numbers to be a triangle of generalized Stirling numbers, to the case of the $q,r$-poly Stirling numbers, which are $q$-analogues of the restricted Stirling numbers defined by Broder and having a polynomial value appearing in their defining recursion. There are two ways to do this and these ways are related by a nice identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06674 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds Komatsu, Takao Bagno, Eli Garber, David Combinatorics Primary: 05A15, Secondary: 05A18, 05A19, 05A30, 11B73 The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion-exclusion principle, and also exponential generating functions. These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized using results of Comtet and Lancaster. We generalize these theorems, which present equivalent conditions for a triangle of numbers to be a triangle of generalized Stirling numbers, to the case of the $q,r$-poly Stirling numbers, which are $q$-analogues of the restricted Stirling numbers defined by Broder and having a polynomial value appearing in their defining recursion. There are two ways to do this and these ways are related by a nice identity. |
| title | Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds |
| topic | Combinatorics Primary: 05A15, Secondary: 05A18, 05A19, 05A30, 11B73 |
| url | https://arxiv.org/abs/2209.06674 |