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

Abstract

In this paper we consider the zeros of the Bergman kernel of the Fock-Bargmann-Hartogs domain {(z, ζ) ∈ C n × C m ; ||ζ|| 2 < e −µ||z|| 2 } where µ > 0. We describe how the existence of zeros of the Bergman kernel depends on the integers m and n with the help of the interlacing property.2000 Mathematics Subject Classification. 32A25.

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“…Since it was first proved by F. Forelli and W. Rudin [7] for the unit disk D = D and ϕ(z) = 1 − |z| 2 , it is called the Forelli-Rudin construction. This formula plays a critical role in establishing explicit formulas of the Bergman kernels of some Hartogs domains (see [21], [22] and [23]). It is also effective in the study of the presence or absence of zeroes of the Bergman kernel, which is called the Lu Qi-Keng problem (see [3] and [24]).…”

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