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

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

2011-05-31

arxiv.org/abs/1106.0036v1

STUDIES IN APPLIED MATHEMATICS 128 (2011) 252--274

Mathematics

Classical Analysis and ODEs

22 pages

Scientific paper

10.1111/j.1467-9590.2011.00541.x

In this paper matrix orthogonal polynomials in the real line are described in terms of a Riemann--Hilbert 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 recursion coefficients to quartic Freud matrix orthogonal polynomials or not.

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