In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair $(G,K)$ of rank one, as a generalization of earlier work by Koornwinder  and subsequently by Koelink, van Pruijssen and Roman ,  for the pair (SU(2)$\times$SU(2), SU(2)), and by Grünbaum, Pacharoni and Tirao  for the pair (SU(3), U(2)). Our method is based on representation theory using an explicit determination of the relevant branching rules. Our matrix valued orthogonal polynomials have the Sturm-Liouville property of being eigenfunctions of a second order matrix valued linear differential operator coming from the Casimir operator, and in fact are eigenfunctions of a commutative algebra of matrix valued linear differential operators coming from the $K$-invariant elements in the universal enveloping algebra of the Lie algebra of $G$.
"Matrix valued orthogonal polynomials for Gelfand pairs of rank one." Tohoku Math. J. (2) 68 (3) 407 - 437, 2016. https://doi.org/10.2748/tmj/1474652266