Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
Mathematical Physics
2014-04-23 v2 Classical Analysis and ODEs
math.MP
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 Painleve' IV differential equation for each family.
Cite
@article{arxiv.1301.2116,
title = {Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials},
author = {Mattia Cafasso and Manuel D. de la Iglesia},
journal= {arXiv preprint arXiv:1301.2116},
year = {2014}
}
Comments
Final version, accepted for publication on CMP: Communications in Mathematical Physics. 24 pages, 1 figure