Bergman Kernels of Elementary Reinhardt Domains

TL;DR

This paper investigates the Bergman kernels of elementary Reinhardt domains in complex space, providing explicit formulas for some and series representations for others, revealing their rational or non-rational nature.

Contribution

It offers explicit computations and series representations of Bergman kernels for elementary Reinhardt domains, extending classical results and highlighting their rationality properties.

Findings

Explicit kernel formulas for certain domains

Identification of non-rational kernels in some cases

Series representations for general domains

Abstract

We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series.