Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalizatio...

Descripción completa

Detalles Bibliográficos
Autores: Durán Guardeño, Antonio José, Grünbaum, Francisco Alberto
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2008
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/182529
Acceso en línea:https://hdl.handle.net/11441/182529
https://doi.org/10.1016/j.jmaa.2008.12.019
Access Level:acceso abierto
Palabra clave:Orthogonal polynomials
Orthogonal matrix polynomials
Recurrence relations
Differential equations
Bispectral problem
id ES_0120d8f4cdd20b3eb45b2c541de933ff
oai_identifier_str oai:idus.us.es:11441/182529
network_acronym_str ES
network_name_str España
repository_id_str
spelling Matrix differential equations and scalar polynomials satisfying higher order recursionsDurán Guardeño, Antonio JoséGrünbaum, Francisco AlbertoOrthogonal polynomialsOrthogonal matrix polynomialsRecurrence relationsDifferential equationsBispectral problemWe show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83–109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88–112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261–280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359–392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.ElsevierAnálisis MatemáticoFQM262: Teoría de la Aproximación2008info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/182529https://doi.org/10.1016/j.jmaa.2008.12.019reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of Mathematical Analysis and Applications, 354 (1), 1-11.10.1016/j.jmaa.2008.12.019info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1825292026-06-17T12:51:07Z
dc.title.none.fl_str_mv Matrix differential equations and scalar polynomials satisfying higher order recursions
title Matrix differential equations and scalar polynomials satisfying higher order recursions
spellingShingle Matrix differential equations and scalar polynomials satisfying higher order recursions
Durán Guardeño, Antonio José
Orthogonal polynomials
Orthogonal matrix polynomials
Recurrence relations
Differential equations
Bispectral problem
title_short Matrix differential equations and scalar polynomials satisfying higher order recursions
title_full Matrix differential equations and scalar polynomials satisfying higher order recursions
title_fullStr Matrix differential equations and scalar polynomials satisfying higher order recursions
title_full_unstemmed Matrix differential equations and scalar polynomials satisfying higher order recursions
title_sort Matrix differential equations and scalar polynomials satisfying higher order recursions
dc.creator.none.fl_str_mv Durán Guardeño, Antonio José
Grünbaum, Francisco Alberto
author Durán Guardeño, Antonio José
author_facet Durán Guardeño, Antonio José
Grünbaum, Francisco Alberto
author_role author
author2 Grünbaum, Francisco Alberto
author2_role author
dc.contributor.none.fl_str_mv Análisis Matemático
FQM262: Teoría de la Aproximación
dc.subject.none.fl_str_mv Orthogonal polynomials
Orthogonal matrix polynomials
Recurrence relations
Differential equations
Bispectral problem
topic Orthogonal polynomials
Orthogonal matrix polynomials
Recurrence relations
Differential equations
Bispectral problem
description We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83–109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88–112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261–280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359–392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.
publishDate 2008
dc.date.none.fl_str_mv 2008
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/182529
https://doi.org/10.1016/j.jmaa.2008.12.019
url https://hdl.handle.net/11441/182529
https://doi.org/10.1016/j.jmaa.2008.12.019
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Mathematical Analysis and Applications, 354 (1), 1-11.
10.1016/j.jmaa.2008.12.019
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869402559490293760
score 15,812455

Similares en LA Referencia