Spherical functions associated with the three dimensional sphere

In this paper, we determine all irreducible spherical functions of any K-type associated to the pair (G;K) = (SO(4); SO(3)). This is accomplished by associating to a vector valued function H = H(u) of a real variable u, which is analytic at u = 0 and whose components are solutions of two coupl...

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Autores principales: Pacharoni, María Inés, Zurrián, Ignacio Nahuel, Tirao, Juan Alfredo
Formato: article
Lenguaje:Inglés
Publicado: 2022
Materias:
Matrix valued spherical functions
Matrix orthogonal polynomials
Three dimensional sphere
The matrix hypergeometric operator
Acceso en línea:http://hdl.handle.net/11086/27866
https://doi.org/10.48550/arXiv.1203.4275
Aporte de:
Repositorio Digital Universitario (UNC) de Universidad Nacional de Córdoba
id I10-R141-11086-27866
record_format dspace
institution Universidad Nacional de Córdoba
institution_str I-10
repository_str R-141
collection Repositorio Digital Universitario (UNC)
language Inglés
topic Matrix valued spherical functions
Matrix orthogonal polynomials
Three dimensional sphere
The matrix hypergeometric operator
spellingShingle Matrix valued spherical functions
Matrix orthogonal polynomials
Three dimensional sphere
The matrix hypergeometric operator
Pacharoni, María Inés
Zurrián, Ignacio Nahuel
Tirao, Juan Alfredo
Spherical functions associated with the three dimensional sphere
topic_facet Matrix valued spherical functions
Matrix orthogonal polynomials
Three dimensional sphere
The matrix hypergeometric operator
description In this paper, we determine all irreducible spherical functions of any K-type associated to the pair (G;K) = (SO(4); SO(3)). This is accomplished by associating to a vector valued function H = H(u) of a real variable u, which is analytic at u = 0 and whose components are solutions of two coupled systems of ordinary dierential equations. By an appropriate conjugation involving Hahn polynomials we uncouple one of the systems. Then this is taken to an uncoupled system of hypergeometric equations, leading to a vector valued solution P = P(u), whose entries are Gegenbauer´s polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of SO(4) to characterize all irreducible spherical functions. The functions P = P(u) corresponding to the irreducible spherical functions of a xed K-type ` are appropriately packaged into a sequence of matrix valued polynomials (Pw)w0 of size (`+1)(`+1). Finally we prove that e Pw = P0 􀀀1Pw is a sequence of matrix orthogonal polynomials with respect to a weight matrix W. Moreover, we show that W admits a second order symmetric hypergeometric operator eD and a rst order symmetric dierential operator e E.
format article
author Pacharoni, María Inés
Zurrián, Ignacio Nahuel
Tirao, Juan Alfredo
author_facet Pacharoni, María Inés
Zurrián, Ignacio Nahuel
Tirao, Juan Alfredo
author_sort Pacharoni, María Inés
title Spherical functions associated with the three dimensional sphere
title_short Spherical functions associated with the three dimensional sphere
title_full Spherical functions associated with the three dimensional sphere
title_fullStr Spherical functions associated with the three dimensional sphere
title_full_unstemmed Spherical functions associated with the three dimensional sphere
title_sort spherical functions associated with the three dimensional sphere
publishDate 2022
url http://hdl.handle.net/11086/27866
https://doi.org/10.48550/arXiv.1203.4275
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AT zurrianignacionahuel sphericalfunctionsassociatedwiththethreedimensionalsphere
AT tiraojuanalfredo sphericalfunctionsassociatedwiththethreedimensionalsphere
bdutipo_str Repositorios
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