M-harmonic reproducing kernels on the ball

TL;DR

This paper derives explicit formulas for the Szeg"o kernel of M-harmonic functions on the complex unit ball using spherical harmonics, revealing the complexity of weighted Bergman kernels and their potential lack of closed-form expressions.

Contribution

It provides explicit hypergeometric function formulas for the M-harmonic Szeg"o kernel and demonstrates the probable absence of closed-form formulas for weighted Bergman kernels.

Findings

Explicit formula for M-harmonic Szeg"o kernel in terms of hypergeometric functions

Most likely no closed-form for weighted Bergman kernels

Utilizes unitary spherical harmonics machinery

Abstract

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant Laplacian on the unit ball of the complex $n$-space. This yields, among others, an explicit formula for the $M$-harmonic Szeg\"o kernel in terms of multivariable as well as single-variable hypergeometric functions, and also shows that most likely there is no explicit (``closed'') formula for the corresponding weighted Bergman kernels.