On spherical expansions of smooth $\gr{SU}{n}$-zonal functions on the unit sphere in $\NC^n$

TL;DR

This paper introduces a new method for constructing spherical harmonic expansions of smooth zonal functions on the complex unit sphere, providing explicit formulas and verifying results for the Poisson–Szeg"o kernel.

Contribution

It presents a novel approach to spherical harmonic expansions on complex spheres, including explicit coefficient formulas and validation against existing results.

Findings

Derived a new formula for expansion coefficients of smooth zonal functions.

Confirmed the expansion for the Poisson–Szeg"o kernel matches Folland's results.

Disproved recent incorrect results in the literature.

Abstract

We give a self-contained presentation of a novel approach to a construction of spherical harmonic expansions on the unit sphere in $\NC^n$. We derive a new formula for coefficients of the expansion of a smooth zonal function defined on the unit sphere and apply it in some special cases. The expansion for the Poisson--Szeg\"o kernel for the unit ball in $\NC^n$ obtained by our method coincides with the result obtained originally by G. Folland, and on the other hand disproves results recently presented in a paper of V.A. Menegatto et al..