The Toda and Painlev\'e Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type

TL;DR

This paper extends the connection between deformed Laguerre polynomials and Painlevé equations to semiclassical matrix-valued orthogonal polynomials, revealing similar integrable structures.

Contribution

It demonstrates that semiclassical matrix-valued Laguerre-type orthogonal polynomials satisfy differential equations analogous to Painlevé III, generalizing previous scalar results.

Findings

Deformation of matrix-valued Laguerre polynomials relates to Painlevé III.

Differential/difference equations govern recursion coefficients.

Results extend scalar orthogonal polynomial theory to matrix-valued case.

Abstract

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, arXiv:0808.3590] 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 semiclassical matrix-valued orthogonal polynomials of Laguerre type.