Explicit formulas of the Bergman kernel for some Reinhardt domains

TL;DR

This paper derives explicit formulas for the Bergman kernel functions on specific Reinhardt domains using hypergeometric functions, advancing the understanding of complex analysis on these domains.

Contribution

It provides closed-form expressions for Bergman kernels on certain Reinhardt domains, utilizing hypergeometric functions for the first time in this context.

Findings

Explicit Bergman kernel formulas for two Reinhardt domains

Application of hypergeometric functions to kernel computation

Enhanced understanding of domain-specific complex analysis

Abstract

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains $\{|z_3|^{\lambda} < |z_1|^{2p} + |z_2|^2, \quad |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}$ and $\{|z_4|^{\lambda} < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \quad (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}$.