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

Manuel D. de La Iglesia
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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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hal-01695474 , version 1 (29-01-2018)

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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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