Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization
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 fr...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147659 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization / F.A. Grünbaum, M.D. de la Iglesia, A. Martínez-Finkelshtein // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 52 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862714121385934848 |
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| author | Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. |
| author_facet | Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. |
| citation_txt | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization / F.A. Grünbaum, M.D. de la Iglesia, A. Martínez-Finkelshtein // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 52 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-07T17:49:29Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147659 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:49:29Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. 2019-02-15T17:09:49Z 2019-02-15T17:09:49Z 2011 Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization / F.A. Grünbaum, M.D. de la Iglesia, A. Martínez-Finkelshtein // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 52 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42C05; 35Q15 https://nasplib.isofts.kiev.ua/handle/123456789/147659 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. The authors are grateful for the excellent job of the referees, whose suggestions and remarks improved the final text.
 The research of the first author was supported in part by the Applied Math. Sciences subprogram of the Of fice of Energy Research, USDOE, under Contract DE-AC03-76SF00098.
 The work of the second author is partially supported by the research project MTM2009-
 12740-C03-02 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF), by Junta de Andaluc´ıa grant FQM-262, by K.U. Leuven research grant OT/04/21, and by Subprograma de estancias de movilidad posdoctoral en el extranjero, MICINN, ref. -2008-0207.
 The third author is supported in part by Junta de Andaluc´ıa grant FQM-229, and by the
 research project MTM2008-06689-C02-01 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF).
 Both the second and the third authors gratefully acknowledge also the support of Junta de
 Andaluc´ıa via the Excellence Research Grant P09-FQM-4643. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization Article published earlier |
| spellingShingle | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. |
| title | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization |
| title_full | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization |
| title_fullStr | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization |
| title_full_unstemmed | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization |
| title_short | Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization |
| title_sort | properties of matrix orthogonal polynomials via their riemann-hilbert characterization |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147659 |
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