Explicit formulae for generalized Stirling and Eulerian numbers
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913330191400960 |
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| author | Küstner, Josef |
| author_facet | Küstner, Josef |
| contents | In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions, complementary symmetric functions, $r$-Whitney numbers and elliptic analogues of rook, Stirling and Lah numbers. Furthermore, we generalize Carlitz' $q$-Eulerian numbers to a Lagrange polynomial extension. We define them by generalizing Worpitzky's identity appropriately, and derive a recursion and an explicit sum formulae. Special cases include $r$-Whitney Eulerian numbers and elliptic Eulerian numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit formulae for generalized Stirling and Eulerian numbers Küstner, Josef Combinatorics Primary 05A10, Secondary 05A30, 05E05, 05A15, 11B65, 11B83, 11B73, 33E05 In this article we generalize the $q$-difference operator due to Carlitz in order to derive explicit sum formulae for several extensions of Stirling numbers of the second kind, including complete homogeneous symmetric functions, complementary symmetric functions, $r$-Whitney numbers and elliptic analogues of rook, Stirling and Lah numbers. Furthermore, we generalize Carlitz' $q$-Eulerian numbers to a Lagrange polynomial extension. We define them by generalizing Worpitzky's identity appropriately, and derive a recursion and an explicit sum formulae. Special cases include $r$-Whitney Eulerian numbers and elliptic Eulerian numbers. |
| title | Explicit formulae for generalized Stirling and Eulerian numbers |
| topic | Combinatorics Primary 05A10, Secondary 05A30, 05E05, 05A15, 11B65, 11B83, 11B73, 33E05 |
| url | https://arxiv.org/abs/2404.16982 |