Relationships Between the Bergman Kernels of Hartogs Domains and Their Base

TL;DR

This paper investigates how the Bergman kernel of a Hartogs domain relates to weighted Bergman kernels of its base domain, providing series representations and transformation formulas to deepen understanding of their connections.

Contribution

It introduces a novel series representation and transformation formula linking the Bergman kernel of Hartogs domains to weighted kernels on the base, advancing kernel analysis techniques.

Findings

Derived a series representation of the Bergman kernel for Hartogs domains.

Established a transformation formula for weighted Bergman kernels.

Presented new relationships connecting kernels of Hartogs domains and their bases.

Abstract

We explore the relationship between the Bergman kernel of a Hartogs domain and weighted Bergman kernels over its base domain. In particular we develop a representation of the Bergman kernel of a Hartogs domain as a series involving weighted Bergman kernels over its base, as well as a "transformation" formula for some weighted Bergman kernels. Other relationships of this type are presented.