Spherical functions associated to the three dimesional sphere

I PACHARONI, J. TIRAO, I. ZURRIÁN

ANNALI DI MATEMATICA PURA ED APPLICATA

SPRINGER HEIDELBERG

Lugar: HEIDELBERG; Año: 2013 p. 1 - 52

0373-3114

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 two coupled systems of ordinary differential 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 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 prove 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 show that $W$ admits a second order symmetric hypergeometric operator $widetilde D$ and a first order symmetric differential operator $widetilde E$.