A variation of the Morris constant term identity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917781928148992 |
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| author | Xin, Guoce Zhang, Chen |
| author_facet | Xin, Guoce Zhang, Chen |
| contents | Morris constant term identity is important due to its equivalence with the well-known Selberg integral. We find a variation of the Morris constant term, denoted $h_n(t)$, in the study of the Ehrhart polynomial $H_n(t)$ of the $n$-th Birkhoff polytope, which consists of all doubly stochastic matrices of order $n$. The constant term $h_n(t)$ corresponds to a particular constant term in the study of $H_n(t)$. We give a characterization of $h_n(t)$ as a polynomial of degree $(n-1)^2$ with additional nice properties involving the Morris constant term identity. We also construct a recursion of $h_n(t)$ using a similar technique for the proof of the Morris constant term identity by Baldoni-Silva and Vergne, and by Xin. This method is applicable to any integer $n\geq 3$. We have produced explicit formulas of $h_n(t)$ for $3 \le n \le 29$ without difficulty. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14356 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A variation of the Morris constant term identity Xin, Guoce Zhang, Chen Combinatorics Primary 05-08, Secondary 03D80, 05A19, 05E05 Morris constant term identity is important due to its equivalence with the well-known Selberg integral. We find a variation of the Morris constant term, denoted $h_n(t)$, in the study of the Ehrhart polynomial $H_n(t)$ of the $n$-th Birkhoff polytope, which consists of all doubly stochastic matrices of order $n$. The constant term $h_n(t)$ corresponds to a particular constant term in the study of $H_n(t)$. We give a characterization of $h_n(t)$ as a polynomial of degree $(n-1)^2$ with additional nice properties involving the Morris constant term identity. We also construct a recursion of $h_n(t)$ using a similar technique for the proof of the Morris constant term identity by Baldoni-Silva and Vergne, and by Xin. This method is applicable to any integer $n\geq 3$. We have produced explicit formulas of $h_n(t)$ for $3 \le n \le 29$ without difficulty. |
| title | A variation of the Morris constant term identity |
| topic | Combinatorics Primary 05-08, Secondary 03D80, 05A19, 05E05 |
| url | https://arxiv.org/abs/2409.14356 |