Matrix-valued Gegenbauer-type polynomials

Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.

Detalles Bibliográficos
Autores principales: Koelink, Erik, De los Ríos, Ana M., Román, Pablo Manuel
Otros Autores: https://orcid.org/0000-0002-2791-385X
Formato: publishedVersion article
Lenguaje:Inglés
Publicado: 2025
Materias:
Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
Acceso en línea:http://hdl.handle.net/11086/555133
https://doi.org/10.1007/s00365-017-9384-4
Aporte de:
Repositorio Digital Universitario (UNC) de Universidad Nacional de Córdoba
id I10-R141-11086-555133
record_format dspace
spelling I10-R141-11086-5551332025-03-14T12:06:58Z Matrix-valued Gegenbauer-type polynomials Koelink, Erik De los Ríos, Ana M. Román, Pablo Manuel https://orcid.org/0000-0002-2791-385X Matrix-valued orthogonal polynomials Matrix-valued differential operators Gegenbauer polynomials Shift operator Darboux factorization info:eu-repo/semantics/publishedVersion Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands. Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España. Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina. Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν > 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν + 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauer-type polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν = 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations. info:eu-repo/semantics/publishedVersion Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands. Fil: De los Ríos, Ana M. Universidad de Sevilla. Facultad de Matemáticas. Departamento de Análisis Matemático; España. Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina. Fil: Román, Pablo Manuel. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Fil: Román, Pablo Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro de Investigación y Estudios de Matemática; Argentina. Matemática Pura 2025-03-13T15:08:08Z 2025-03-13T15:08:08Z 2017 article Koelink, E., De los Ríos, A. M. y Román, P. M. (2017). Matrix-valued Gegenbauer-type polynomials. Constructive Approximation, 46 (3), 459-487. https://doi.org/10.1007/s00365-017-9384-4 0176-4276 http://hdl.handle.net/11086/555133 1432-0940 https://doi.org/10.1007/s00365-017-9384-4 eng Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ Impreso; Electrónico y/o Digital
institution Universidad Nacional de Córdoba
institution_str I-10
repository_str R-141
collection Repositorio Digital Universitario (UNC)
language Inglés
topic Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
spellingShingle Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
Koelink, Erik
De los Ríos, Ana M.
Román, Pablo Manuel
Matrix-valued Gegenbauer-type polynomials
topic_facet Matrix-valued orthogonal polynomials
Matrix-valued differential operators
Gegenbauer polynomials
Shift operator
Darboux factorization
description Fil: Koelink, Erik. Radboud Universiteit. Institute for Mathematics, Astrophysics and Particle Physics; Netherlands.
author2 https://orcid.org/0000-0002-2791-385X
author_facet https://orcid.org/0000-0002-2791-385X
Koelink, Erik
De los Ríos, Ana M.
Román, Pablo Manuel
format publishedVersion
article
author Koelink, Erik
De los Ríos, Ana M.
Román, Pablo Manuel
author_sort Koelink, Erik
title Matrix-valued Gegenbauer-type polynomials
title_short Matrix-valued Gegenbauer-type polynomials
title_full Matrix-valued Gegenbauer-type polynomials
title_fullStr Matrix-valued Gegenbauer-type polynomials
title_full_unstemmed Matrix-valued Gegenbauer-type polynomials
title_sort matrix-valued gegenbauer-type polynomials
publishDate 2025
url http://hdl.handle.net/11086/555133
https://doi.org/10.1007/s00365-017-9384-4
work_keys_str_mv AT koelinkerik matrixvaluedgegenbauertypepolynomials
AT delosriosanam matrixvaluedgegenbauertypepolynomials
AT romanpablomanuel matrixvaluedgegenbauertypepolynomials
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