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

In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=(\mathrm{S}\mathrm{O}(n+1),\mathrm{S}\mathrm{O}(n))$ 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 $D$.