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.

CP - 2 PB - Springer Verlag CY - Berlin ; Heildeberg VL - 326 EP - 559-583 N1 - Mars 2014 L1 - eng L4 - 1 U4 - http://dx.doi.org/10.1007/s00220-013-1853-4 J1 - Commun. Math. Phys. L2 - 10.1007/s00220-013-1853-4 ER -