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The Toda and Painlev\'e systems associated with semiclassical matrix-valued orthogonal polynomials of Laguerre type

Abstract : Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In "Painlev\'e III and a singular linear statistics in Hermitian random matrix ensembles, I", the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlev\'e III equation and its B\"acklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of matrix-valued orthogonal polynomials of Laguerre type.
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https://hal.archives-ouvertes.fr/hal-01695474
Contributor : Mattia Cafasso <>
Submitted on : Monday, January 29, 2018 - 1:48:07 PM
Last modification on : Monday, March 9, 2020 - 6:15:57 PM

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  • HAL Id : hal-01695474, version 1
  • ARXIV : 1801.08740

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Mattia Cafasso, Manuel D. de la Iglesia. The Toda and Painlev\'e systems associated with semiclassical matrix-valued orthogonal polynomials of Laguerre type. 2018. ⟨hal-01695474⟩

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