Spherical Functions of Fundamental $K$-Types Associated with the $n$-Dimensional Sphere

Abstract

In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=(\mathrm{SO}(n+1),\mathrm{SO}(n))$ in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same $K$-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$ supported on $[0,1]$. Moreover, we show that $W$ has a second order symmetric hypergeometric operator $D$.

References in corpus (1)

• Matrix Valued Orthogonal Polynomials for Gelfand Pairs of Rank One

Cited by in corpus (6)

• Time and Band Limiting for Matrix Valued Functions, an Example
• Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
• Time and band limiting for matrix valued functions: an integral and a commuting differential operator
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs
• Pre-sequences of matrix orthogonal polynomials