On a class of elliptic orthogonal polynomials and their integrability
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910427789656064 |
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| author | Desiraju, Harini Latimer, Tomas Lasic Roffelsen, Pieter |
| author_facet | Desiraju, Harini Latimer, Tomas Lasic Roffelsen, Pieter |
| contents | Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a constant weight function, and use the Riemann-Hilbert problem to derive recurrence relations and differential equations for the orthogonal polynomials. We further show that the sub-class of even polynomials is associated to the elliptic form of Painlevé VI, with the tau function given by the Hankel determinant of even moments, up to a scaling factor. The first iteration of these even polynomials relates to the special case of Painlevé VI studied by Hitchin in relation to self-dual Einstein metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04404 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a class of elliptic orthogonal polynomials and their integrability Desiraju, Harini Latimer, Tomas Lasic Roffelsen, Pieter Classical Analysis and ODEs Mathematical Physics Exactly Solvable and Integrable Systems Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a constant weight function, and use the Riemann-Hilbert problem to derive recurrence relations and differential equations for the orthogonal polynomials. We further show that the sub-class of even polynomials is associated to the elliptic form of Painlevé VI, with the tau function given by the Hankel determinant of even moments, up to a scaling factor. The first iteration of these even polynomials relates to the special case of Painlevé VI studied by Hitchin in relation to self-dual Einstein metrics. |
| title | On a class of elliptic orthogonal polynomials and their integrability |
| topic | Classical Analysis and ODEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2305.04404 |