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 ) ∈ C 2 : |z 1 | 4/q1 + |z 2 | 4/q2 < 1} for any positive integers q 1 and q 2 . We also prove that among the domains Dp:= {(z 1 , z 2 ) ∈ C 2 : |z 1 | 2p1 + |z 2 | 2p2 < 1} in C 2 with p = (p 1 , p 2 ) ∈ N 2 , the Bergman kernel is represented in terms of closed forms if and only if p = (p1, 1), (1, p2), or p (2, 2).