Hankel Determinants for a Class of Weighted Lattice Paths
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914958569111552 |
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| author | Wang, Ying Zhang, Zihao |
| author_facet | Wang, Ying Zhang, Zihao |
| contents | In this paper, our primary goal is to calculate the Hankel determinants for a class of lattice paths, which are distinguished by the step set consisting of \(\{(1,0), (2,0), (k-1,1), (-1,1)\}\), where the parameter \(k\geq 4\). These paths are constrained to return to the $x$-axis and remain above the \(x\)-axis.
When calculating for \(k = 4\), the problem essentially reduces to determining the Hankel determinant of \(E(x)\), where \(E(x)\) is defined as \[ E(x) = \frac{a}{E(x)x^2(dx^2 - bx - 1) + cx^2 + bx + 1}. \] Our approach involves employing the Sulanke-Xin continued fraction transform to derive a set of recurrence relations, which in turn yield the desired results. For \(k \geq 5\), we utilize a class of shifted periodic continued fractions as defined by Wang-Xin-Zhai, thereby obtaining the results presented in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hankel Determinants for a Class of Weighted Lattice Paths Wang, Ying Zhang, Zihao Combinatorics Primary 15A15, Secondary 05A15, 11B83 In this paper, our primary goal is to calculate the Hankel determinants for a class of lattice paths, which are distinguished by the step set consisting of \(\{(1,0), (2,0), (k-1,1), (-1,1)\}\), where the parameter \(k\geq 4\). These paths are constrained to return to the $x$-axis and remain above the \(x\)-axis. When calculating for \(k = 4\), the problem essentially reduces to determining the Hankel determinant of \(E(x)\), where \(E(x)\) is defined as \[ E(x) = \frac{a}{E(x)x^2(dx^2 - bx - 1) + cx^2 + bx + 1}. \] Our approach involves employing the Sulanke-Xin continued fraction transform to derive a set of recurrence relations, which in turn yield the desired results. For \(k \geq 5\), we utilize a class of shifted periodic continued fractions as defined by Wang-Xin-Zhai, thereby obtaining the results presented in this paper. |
| title | Hankel Determinants for a Class of Weighted Lattice Paths |
| topic | Combinatorics Primary 15A15, Secondary 05A15, 11B83 |
| url | https://arxiv.org/abs/2409.18609 |