The Bergman Kernel of Complex Ovals and Multivariable Hypergeometric Functions

Abstract

We compute explicitly the Bergman and Szego kernels for a class of pseudoconvex domains. The kernels are expressed in terms of Appell's multivariable hypergeometric functions. These explicit formulas are applied to investigate the asymptotic behavior of the Bergman kernel near some weakly pseudoconvex boundary points.

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

“…The Bergman kernel for D p was obtained in [7,8], and similarly we can get the Szegő kernel. p j be a positive integer for j = 1, . .…”

“…If D is a homogeneous domain in C n , then Hua [13] and Yin [17] computed the explicit formula of the Bergman kernel for classical domains and exceptional domains, respectively. Recently many mathematicians [5,6,7,8,9,10,11,18] have made efforts to find the explicit formulas of the Bergman kernels for nonhomogeneous domains.…”

“…Similarly we also have to compute F 2 (γ 1 + γ 2 ; 1, 1; γ 1 , γ 2 ; x, y) for the Szegő kernel. One can see the following transformation formula in [8].…”

“…For example, using Bell's transformation formula of the Bergman kernel under the proper holomorphic mappings, the Bergman kernel for D (1/q 1 ,1/q 2 ) for (q 1 , q 2 ) ∈ N 2 is obtained as a finite sum in [8] and [5].…”

Proper holomorphic liftings and new formulas for the Bergman and Szegő kernels

“…. × M m d ,n d (C) which contains the generalized oval domains considered in [D'A1], [D'A2], [FH1] and [FH2] and the minimal ball introduced in [HP]. We compute their Bergman and Szegő kernels.…”