Spherical functions associated with the three dimensional sphere

Abstract

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.