CIEM   05476
CENTRO DE INVESTIGACION Y ESTUDIOS DE MATEMATICA
Unidad Ejecutora - UE
artículos
Título:
Spherical functions to the three dimensional sphere
Autor/es:
INÉS PACHARONI; JUAN A. TIRAO; IGNACIO ZURRIÁN
Revista:
ANNALI DI MATEMATICA PURA ED APPLICATA
Editorial:
SPRINGER HEIDELBERG
Referencias:
Lugar: HEIDELBERG; Año: 2012
ISSN:
0373-3114
Resumen:
In this paper, we determine all irreducible spherical functions$Phi$ of any $K $-type associated to the pair$(G,K)=(SO(4),SO(3))$. This is accomplished by associating to$Phi$ a vector valued function $H=H(u)$ of a real variable $u$,which is analytic at $u=0$ and whose components are solutions of twocoupled systems of ordinary differential equations. By anappropriate conjugation involving Hahn polynomials we uncouple oneof 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 fixed $K$-type $pi_ell$ are appropriately packaged into a sequence
of matrix valued polynomials $(P_w)_{wge0}$ of size $(ell+1)
imes(ell+1)$. Finally we proved that $widetilde P_w={P_0}^{-1}P_w$ is a
sequence of matrix orthogonal polynomials with respect to a weight
matrix $W$. Moreover we showed that $W$ admits a second order symmetric
hypergeometric operator $widetilde D$ and a first order symmetric
differential operator $widetilde E$.