THE BERGMAN KERNEL FOR INTERSECTION OF TWO COMPLEX ELLIPSOIDS

Tomasz Beberok
2016 Bulletin of the Korean Mathematical Society  
In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for intersection of two complex ellipsoids {z ∈ C 3 : |z 1 | p + |z 2 | q < 1, |z 1 | p + |z 3 | r < 1}. We consider cases p = 6, q = r = 2 and p = q = r = 2. We also investigate the Lu Qi-Keng problem for p = q = r = 2. in [18] compute Bergman kernel for nonhomogeneous domain for any positive integers q 1 and q 2 . The goal of this paper is
more » ... l of this paper is to give Bergman kernel for {z ∈ C 3 : |z 1 | p + |z 2 | q < 1, |z 1 | p + |z 3 | r < 1} in cases when p = 6, q = r = 2 or p = q = r = 2. Main results The following are the main theorems of this paper. Theorem 2.1. The Bergman kernel for D 1 = {(z 1 , z 2 , z 3 ) ∈ C 3 : |z 1 | 2 +|z 2 | 2 < 1, |z 1 | 2 + |z 3 | 2 < 1} is given by where ν i = z i w i for i = 1, 2, 3. Theorem 2.2. The Bergman kernel for D 2 = {(z 1 , z 2 , z 3 ) ∈ C 3 : |z 1 | 6 +|z 2 | 2 < 1, |z 1 | 6 + |z 3 | 2 < 1} is given by
doi:10.4134/bkms.b150502 fatcat:ceeyk24isre7phphmggu2jaz34