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Communications in Mathematical Physics

, Volume 326, Issue 2, pp 559–583 | Cite as

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

  • Mattia Cafasso
  • Manuel D. de la Iglesia
Article

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.

Keywords

Matrix Polynomial Hilbert Problem Fredholm Determinant Airy Kernel Matrix Orthogonal Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.LUNAM Université, LAREMA, Université d’AngersAngersFrance
  2. 2.Departamento de Análisis MatemáticoUniversidad de SevillaSevillaSpain

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