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arXiv · 2305.04404

On a class of elliptic orthogonal polynomials and their integrability

Harini Desiraju·Tomas Lasic Latimer·Pieter Roffelsen
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Abstract

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 Painlev\'e 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 Painlev\'e VI studied by Hitchin in relation to self-dual Einstein metrics.

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Harini Desiraju, Tomas Lasic Latimer, Pieter Roffelsen. 2023-05-08. On a class of elliptic orthogonal polynomials and their integrability. https://doi.org/10.1007/s00365-024-09687-z

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