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


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.


Matrix Polynomial Hilbert Problem Fredholm Determinant Airy Kernel Matrix Orthogonal Polynomial 
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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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