Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 076, 17 pages      arXiv:1801.08740
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type

Mattia Cafasso a and Manuel D. de la Iglesia b
a) LAREMA - Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers, France
b) Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Mexico City, Mexico

Received March 28, 2018, in final form July 16, 2018; Published online July 21, 2018

Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297] 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 Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Laguerre type.

Key words: Painlevé equations; Toda lattices; Riemann-Hilbert problems; matrix-valued orthogonal polynomials..

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