Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II
- Autores
- Koelink, Erik; Van Pruijssen, Maarten; Román, Pablo Manuel
- Año de publicación
- 2013
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- In a previous paper we have introduced matrixvalued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. In this paper, the matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operators of first and second order of which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials Pn. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group-theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2).
Fil: Koelink, Erik. Radboud Universiteit; Países Bajos
Fil: Van Pruijssen, Maarten. Radboud Universiteit; Países Bajos
Fil: Román, Pablo Manuel. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica. Seccion Matematica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina - Materia
-
Spherical Functions
Matrix Valued Orthogonal Polynomials - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/12273
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Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), IIKoelink, ErikVan Pruijssen, MaartenRomán, Pablo ManuelSpherical FunctionsMatrix Valued Orthogonal Polynomialshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In a previous paper we have introduced matrixvalued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. In this paper, the matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operators of first and second order of which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials Pn. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group-theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2).Fil: Koelink, Erik. Radboud Universiteit; Países BajosFil: Van Pruijssen, Maarten. Radboud Universiteit; Países BajosFil: Román, Pablo Manuel. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica. Seccion Matematica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaKyoto Univ2013-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/12273Koelink, Erik; Van Pruijssen, Maarten; Román, Pablo Manuel; Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II; Kyoto Univ; Publications Of The Research Institute For Mathematical Sciences; 49; 2; 4-2013; 271-3120034-5318enginfo:eu-repo/semantics/altIdentifier/doi/10.4171/PRIMS/106info:eu-repo/semantics/altIdentifier/url/http://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=49&iss=2&rank=3info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:04:48Zoai:ri.conicet.gov.ar:11336/12273instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:04:48.594CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
title |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
spellingShingle |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II Koelink, Erik Spherical Functions Matrix Valued Orthogonal Polynomials |
title_short |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
title_full |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
title_fullStr |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
title_full_unstemmed |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
title_sort |
Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II |
dc.creator.none.fl_str_mv |
Koelink, Erik Van Pruijssen, Maarten Román, Pablo Manuel |
author |
Koelink, Erik |
author_facet |
Koelink, Erik Van Pruijssen, Maarten Román, Pablo Manuel |
author_role |
author |
author2 |
Van Pruijssen, Maarten Román, Pablo Manuel |
author2_role |
author author |
dc.subject.none.fl_str_mv |
Spherical Functions Matrix Valued Orthogonal Polynomials |
topic |
Spherical Functions Matrix Valued Orthogonal Polynomials |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
In a previous paper we have introduced matrixvalued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. In this paper, the matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operators of first and second order of which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials Pn. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group-theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2). Fil: Koelink, Erik. Radboud Universiteit; Países Bajos Fil: Van Pruijssen, Maarten. Radboud Universiteit; Países Bajos Fil: Román, Pablo Manuel. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica. Seccion Matematica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina |
description |
In a previous paper we have introduced matrixvalued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. In this paper, the matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P_n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operators of first and second order of which the P_n's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials Pn. These differential operators are also crucial in expressing the matrix entries of P_nL as a product of a Racah and a Gegenbauer polynomial. We also present a group-theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2). |
publishDate |
2013 |
dc.date.none.fl_str_mv |
2013-04 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
format |
article |
status_str |
publishedVersion |
dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11336/12273 Koelink, Erik; Van Pruijssen, Maarten; Román, Pablo Manuel; Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II; Kyoto Univ; Publications Of The Research Institute For Mathematical Sciences; 49; 2; 4-2013; 271-312 0034-5318 |
url |
http://hdl.handle.net/11336/12273 |
identifier_str_mv |
Koelink, Erik; Van Pruijssen, Maarten; Román, Pablo Manuel; Matrix-valued Orthogonal Polynomials Related to (SU(2) × SU(2), SU(2)), II; Kyoto Univ; Publications Of The Research Institute For Mathematical Sciences; 49; 2; 4-2013; 271-312 0034-5318 |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/doi/10.4171/PRIMS/106 info:eu-repo/semantics/altIdentifier/url/http://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=49&iss=2&rank=3 |
dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
eu_rights_str_mv |
openAccess |
rights_invalid_str_mv |
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
dc.format.none.fl_str_mv |
application/pdf application/pdf |
dc.publisher.none.fl_str_mv |
Kyoto Univ |
publisher.none.fl_str_mv |
Kyoto Univ |
dc.source.none.fl_str_mv |
reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
reponame_str |
CONICET Digital (CONICET) |
collection |
CONICET Digital (CONICET) |
instname_str |
Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.name.fl_str_mv |
CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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13.070432 |