New formulas of the Bergman kernels for complex ellipsoids in $\mathbb {C}^2$

Abstract:

We compute the explicit formula of the Bergman kernel for a nonhomogeneous domain $\{(z_1,z_2)\in \mathbb {C}^2:|z_1|^{4/q_1}+|z_2|^{4/q_2}<1\}$ for any positive integers $q_1$ and $q_2$. We also prove that among the domains $D_p:=\{(z_1,z_2)\in \mathbb {C}^2:|z_1|^{2p_1}+|z_2|^{2p_2}<1\}$ in $\mathbb {C}^2$ with $p=(p_1,p_2)\in \mathbb {N}^2$, the Bergman kernel is represented in terms of closed forms if and only if $p=(p_1,1),(1,p_2)$, or $p=(2,2)$.