New Formulas of the Bergman Kernels for Complex Ellipsoids in C

We compute the explicit formula of the Bergman kernel for a nonhomogeneous domain {(z1, z2) ∈ C2 : |z1|1 + |z2|2 < 1} for any positive integers q1 and q2. We also prove that among the domains Dp := {(z1, z2) ∈ C2 : |z1|1 + |z2|2 < 1} in C2 with p = (p1, p2) ∈ N2, the Bergman kernel is represented in terms of closed forms if and only if p = (p1, 1), (1, p2), or p = (2, 2).