On spherical expansions of smooth SU(n)-zonal functions on the unit sphere in CnCn

We give a self-contained presentation of a novel approach to the spherical harmonic expansions of smooth zonal functions defined on the unit sphere in CnCn. The main new result is a formula expressing the coefficients of the expansion in terms of the Taylor coefficients of the profile function. This enables us to give a new form of the classical Funk–Hecke formula for the case of complex spheres. As another application we give a new derivation the spherical harmonic expansion for the Poisson–Szegö kernel for the unit ball in CnCn obtained originally by Folland.