Riemann–Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement

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n this paper, matrix orthogonal polynomials in the real line are described in terms of a RiemannHilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not.
Wiley-Blackwell. The authors thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS2008-00200. GAC acknowledges the support of the grant Universidad Complutense de Madrid. Finally, MM reckons illuminating discussions with Dr. Mattia Cafasso in relation with orthogonality and singularity confinement, and both authors are grateful to Prof. Gabriel Álvarez Galindo for several discussions and for the experimental confirmation, via Mathematica, of the existence of the confinement of singularities in the 2 × 2 case.
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