A remark on the Bergman kernels of the Cartan-Hartogs domains
Let \(R\) be a Cartan domain (bounded circled irreducible homogeneous domain) and \(N\) its generic norm. For a positive integer \(m\) and \(s>0\), the Cartan-Hartogs domain \(\Omega_{R}(m,s)=\Omega_{R}\) is defined by \[ \Omega_{R}(m,s)=\left\{ (z,\zeta)\in R\times\mathbb{C}^{m}\Big|\left\| \zeta\right\| ^{2}<N(z,z)^{s}\right\} . \] It is known that the Bergman kernel of \(\Omega_{R}\) has the form \[ K_{\Omega_{R}}\big( (z,\zeta),(z^{\prime},\zeta^{\prime})\big) =\frac{K_{R}(z,z^{\prime})}{N(z,z^{\prime})^{ms}}F\left( \frac{\left\langle \zeta,\zeta^{\prime}\right\rangle }{N(z,z^{\prime})^{s}}\right), \] where \(K_{R}\) is the Bergman kernel of \(R\) and \(F(t)\) a rational function with a unique pole at \(t=1\) [\textit{W. Yin, K. Lu} and \textit{G. Roos}, Sci.~China, Ser.~A 47, No. 3, 352--371 (2004; Zbl 1079.32002)]. The note gives a new interpretation of the rational function \(F\) as a linear combination of polylogarithms with negative entire parameters. In the last part of the note, sufficient conditions are given for \(\Omega_{R}\) being a Lu Qikeng domain, that is, having a zero-free Bergman kernel. It is shown that, for \(R\) and \(s\) fixed, there exists \(m_{0}\) such that the Bergman kernel of \(\Omega_{R}\) is zero-free for all \(m\geq m_{0}\). The minimal value of \(m_{0}\) was obtained previously by \textit{F. Z. Demmad-Abdessameud} [Rend. Semin. Mat., Univ.~Politec. Torino 67, No. 1, 55--89 (2009; Zbl 1181.32027)] when the base Cartan domain \(R\) has dimension at most \(4\).
scientific article; zbMATH DE number 6017091
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