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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| Datum: | 2011 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2011
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| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/147659 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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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irk-123456789-1476592019-02-16T01:25:39Z Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. 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. 2011 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147659 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. |
| spellingShingle |
Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Grünbaum, F.A. de la Iglesia, M.D. Martínez-Finkelshtein, A. |
| author_sort |
Grünbaum, F.A. |
| title |
Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization |
| title_short |
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_sort |
properties of matrix orthogonal polynomials via their riemann-hilbert characterization |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147659 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-11T02:35:02Z |
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