Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 098, 31 pages      arXiv:1106.1307

Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

F. Alberto Grünbaum a, Manuel D. de la Iglesia b and Andrei Martínez-Finkelshtein c
a) Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720 USA
b) Departamento de Análisis Matemático, Universidad de Sevilla, Apdo (P.O. BOX) 1160, 41080 Sevilla, Spain
c) Departamento de Estadística y Matemática Aplicada, Universidad de Almería, 04120 Almería, Spain

Received June 09, 2011, in final form October 20, 2011; Published online October 25, 2011

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.

Key words: matrix orthogonal polynomials; Riemann-Hilbert problems.

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