Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908528059351040 |
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| author | Crampé, Nicolas Labriet, Quentin Morey, Lucia Tsujimoto, Satoshi Vinet, Luc Zhedanov, Alexei |
| author_facet | Crampé, Nicolas Labriet, Quentin Morey, Lucia Tsujimoto, Satoshi Vinet, Luc Zhedanov, Alexei |
| contents | The rank two Jacobi algebra $\mathcal{J}_2$ is used to provide an interpretation of the two-variable Jacobi polynomials $J_{n,k}^{(a,b,c)}(x,y)$ on the triangle, as overlaps between two representation bases. The subalgebra structure of $\mathcal{J}_2$ depicted via a pentagonal graph is exploited to find the explicit expression of the two-variable functions in terms of univariate Jacobi polynomials. It is also seen to provide an explanation for the fact that the expansion on the basis $J_{n,k}^{(a,b,c)}(x,y)$ of the polynomials obtained from the latter by permuting the variables $x,y, z=1-x-y$ and the parameters $(a,b,c)$ is given in terms of Racah polynomials. The underlying order-three symmetry is discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way Crampé, Nicolas Labriet, Quentin Morey, Lucia Tsujimoto, Satoshi Vinet, Luc Zhedanov, Alexei Representation Theory 33C45, 33C50, 81R05 The rank two Jacobi algebra $\mathcal{J}_2$ is used to provide an interpretation of the two-variable Jacobi polynomials $J_{n,k}^{(a,b,c)}(x,y)$ on the triangle, as overlaps between two representation bases. The subalgebra structure of $\mathcal{J}_2$ depicted via a pentagonal graph is exploited to find the explicit expression of the two-variable functions in terms of univariate Jacobi polynomials. It is also seen to provide an explanation for the fact that the expansion on the basis $J_{n,k}^{(a,b,c)}(x,y)$ of the polynomials obtained from the latter by permuting the variables $x,y, z=1-x-y$ and the parameters $(a,b,c)$ is given in terms of Racah polynomials. The underlying order-three symmetry is discussed. |
| title | Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way |
| topic | Representation Theory 33C45, 33C50, 81R05 |
| url | https://arxiv.org/abs/2509.07949 |