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

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