The Zeros of the Bergman Kernel for Some Reinhardt Domains

Abstract

We consider the Reinhardt domain Dn={(ζ,z)∈C×Cn:|ζ|2<(1-|z1|2)⋯(1-|zn|2)}. We express the explicit closed form of the Bergman kernel for Dn using the exponential generating function for the Stirling number of the second kind. As an application, we show that the Bergman kernel Kn for Dn has zeros if and only if n≥3. The study of the zeros of Kn is reduced to some real polynomial with coefficients which are related to Bernoulli numbers. This result is a complete characterization of the existence of zeros of the Bergman kernel for Dn for all positive integers n.