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Non-Commutative Painlevé Equations and Hermite-Type Matrix Orthogonal Polynomials

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Abstract

We study double integral representations of Christoffel–Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its–Izergin–Korepin–Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painlevé IV differential equation for each family.

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Authors and Affiliations

  1. LUNAM Université, LAREMA, Université d’Angers, 2 Boulevard Lavoisier, 49100, Angers, France

    Mattia Cafasso

  2. Departamento de Análisis Matemático, Universidad de Sevilla, Apdo (P. O. BOX) 1160, 41080, Sevilla, Spain

    Manuel D. de la Iglesia

Authors
  1. Mattia Cafasso
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  2. Manuel D. de la Iglesia
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Correspondence to Manuel D. de la Iglesia.

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Communicated by P. Deift

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Cafasso, M., de la Iglesia, M.D. Non-Commutative Painlevé Equations and Hermite-Type Matrix Orthogonal Polynomials. Commun. Math. Phys. 326, 559–583 (2014). https://doi.org/10.1007/s00220-013-1853-4

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  • DOI: https://doi.org/10.1007/s00220-013-1853-4

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