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This article is cited in 20 scientific papers (total in 20 papers)
Properties of Matrix Orthogonal Polynomials via their Riemann–Hilbert Characterization
F. Alberto Grünbauma, Manuel D. de la Iglesiab, Andrei Martínez-Finkelshteinc 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
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.
Keywords:
matrix orthogonal polynomials, Riemann–Hilbert problems.
Received: June 9, 2011; in final form October 20, 2011; Published online October 25, 2011
Citation:
F. Alberto Grünbaum, Manuel D. de la Iglesia, Andrei Martínez-Finkelshtein, “Properties of Matrix Orthogonal Polynomials via their Riemann–Hilbert Characterization”, SIGMA, 7 (2011), 098, 31 pp.
Linking options:
https://www.mathnet.ru/eng/sigma656 https://www.mathnet.ru/eng/sigma/v7/p98
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Abstract page: | 174 | Full-text PDF : | 45 | References: | 29 |
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