Generating functions for the spherical functions on some classical Gelfand pairs (Q1201499)

It is known that the zonal spherical functions on the unit sphere in real space \(\mathbb{R}^ n\) can be expressed through the Gegenbauer polynomials, and have a generating function which has the form of a power of a trinomial. The purpose of the article is to announce that similar generating functions also exist for the zonal spherical functions in the cases of complex and quaternion spaces \(\mathbb{C}^ n\) and \(\mathbb{H}^ n\).