Spherical functions associated with the three-dimensional sphere

Abstract

In this paper, we determine all irreducible spherical functions ф of any K-type associated with the pair (G,K)=(SO(4),SO(3)). This is accomplished by associating with ф a vector-valued function H=H(u) of a real variable 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 (Formula presented.) are appropriately packaged into a sequence of matrix-valued polynomials (Formula presented.) of size (Formula presented.). Finally, we prove that (Formula presented.) 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 (Formula presented.) and a first-order symmetric differential operator (Formula presented.).