The Bergman kernel of the Fock–Bargmann–Hartogs domain and the polylogarithm function

Abstract

We consider the Fock–Bargmann–Hartogs domain D _n,m which is defined by the inequality where (z, ζ) ∈ ℂ ⁿ  × ℂ ^m and μ > 0. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover, using the interlacing property, we describe how the existence of zeros of the Bergman kernel depends on the integers m and n.

Keywords:

• Bergman kernel
• weighted Bergman kernel
• Fock–Bargmann space
• polylogarithm function
• Lu Qi-Keng problem
• Forelli–Rudin construction

AMS Subject classifications:

• 32A25
• 32A07

Acknowledgements

The author would like to express his sincere gratitude to Professors Hideyuki Ishi and Hiroyuki Ochiai and Dr Daisuke Shiomi for their helpful advice and discussions. The author also acknowledges the encouragement and helpful comments by Professor Takeo Ohsawa on this study.