Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

Abstract

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.

References in corpus (5)

• The Analytic Theory of Matrix Orthogonal Polynomials
• Infinite J-matrices and a matrix moment problem
• Asymptotics for a special solution of the thirty fourth Painleve equation
• A phase transition for non-intersecting Brownian motions, and the Painleve II equation
• Random walks and orthogonal polynomials: some challenges

Cited by in corpus (10)

• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
• Singularity confinement for matrix discrete Painleve Equations
• Reducibility of Matrix Weights
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• Ladder relations for a class of matrix valued orthogonal polynomials
• The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
• Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system
• Wiener-Hopf factorizations and matrix-valued orthogonal polynomials
• Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV