Spherical Functions Associated With the Three Dimensional Sphere

TL;DR

This paper classifies all irreducible spherical functions related to the pair (SO(4), SO(3)), linking them to hypergeometric functions, Gegenbauer polynomials, and matrix orthogonal polynomials, using representation theory and differential equations.

Contribution

It provides a complete characterization of irreducible spherical functions for (SO(4), SO(3)) and constructs associated matrix orthogonal polynomials with explicit differential operators.

Findings

All irreducible spherical functions are characterized via hypergeometric and Gegenbauer polynomials.

The associated matrix polynomials form a sequence of orthogonal polynomials with explicit weight matrices.

The weight matrix admits second order hypergeometric and first order differential operators.

Abstract

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)_{w\ge0} of size (\ell+1)\times(\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.