Spherical Functions of Fundamental $K$-Type on the $n$-dimensional Sphere

JUAN A. TIRAO; IGNACIO N. ZURRIÁN

SYMMETRY, INTEGRABILITY AND GEOMETRY

NATL ACAD SCI UKRAINE

Año: 2014

1815-0659

In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=(SO(n+1),SO(n))$  in terms of  matrix hypergeometricfunctions. The output of these description is that the irreducible spherical functions of the same  $K$-type are encoded  news examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$. Moreover, we show that $W$ admits a second order symmetric hypergeometric operator $D$.