$K$-Finite Matrix Elements of Irreducible Harish-Chandra Modules are Hypergeometric

We show that each $K$-finite matrix element of an irreducible infinite-dimensional representation of a semisimple Lie group can be obtained from spherical functions by a finite collection of operations. In particular, each matrix element admits a finite expression in the terms of the Heckman–Opdam hypergeometric functions.