The Bergman kernel of a certain Hartogs domain and the polylogarithm function

TL;DR

This paper derives an explicit formula for the Bergman kernel of a specific Hartogs domain related to the Fock-Bargmann space using polylogarithm functions and addresses the Lu Qi-Keng problem for this domain.

Contribution

It provides a new explicit formula for the Bergman kernel involving polylogarithms and solves the Lu Qi-Keng problem in certain cases for this domain.

Findings

Explicit Bergman kernel formula in terms of polylogarithms

Solution to the Lu Qi-Keng problem in some cases

Enhanced understanding of the domain's complex geometry

Abstract

We consider a certain Hartogs domain which is related to the Fock-Bargmann space. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover we solve the Lu Qi-Keng problem of the domain in some cases.