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Symmetry, Integrability and Geometry: Methods and Applications, 2011, Volume 7, 098, 31 pp.
DOI: https://doi.org/10.3842/SIGMA.2011.098
(Mi sigma656)
 

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
References:
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
Bibliographic databases:
Document Type: Article
MSC: 42C05; 35Q15
Language: English
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.
Citation in format AMSBIB
\Bibitem{GruDe Mar11}
\by F. Alberto Gr\"unbaum, Manuel D. de la Iglesia, Andrei Mart{\'\i}nez-Finkelshtein
\paper Properties of Matrix Orthogonal Polynomials via their Riemann--Hilbert Characterization
\jour SIGMA
\yr 2011
\vol 7
\papernumber 098
\totalpages 31
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\crossref{https://doi.org/10.3842/SIGMA.2011.098}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2861178}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84857070529}
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  • This publication is cited in the following 20 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Symmetry, Integrability and Geometry: Methods and Applications
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