A note on the Bergman kernel of a certain Hartogs domain
Let \(F\) be a non-increasing continuous function from an interval \((0,B]\) into \((0,+\infty].\) Suppose that \(F\) extends to a meromorphic function on \(\{z\in \mathbb{C} : |z|<B\}.\) Let \[ \Omega_{D_F.m}= \Big\{ (z,\zeta ) \in D_F \times \mathbb{C}^m : |\zeta|^2 <\Big(F\big(|z_1|^2\big)- |z_2|^2\Big)^s\Big\}, \] where \(s>0\) and the domain \(D_F\) is defined by \[ D_F= \big\{ z\in \mathbb{C}^2 : |z_1|^2 <B,\; |z_2|^2 < F(|z_1|^2)\big\}. \] The author gives an explicit formula for the Bergman kernel of \(\Omega_{D_F.m},\) where the so-called polylogarithm function \(Li_{-n}(z) = \sum_{k=1}^\infty k^n z^k\) is involved.
scientific article; zbMATH DE number 6109864