  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) x SU(2),SU(2)), and by Gr\"unbaum,
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 $U(g_c)^K$.
