The Bergman kernels of generalized Bergman–Hartogs domains

We define the generalized Bergman–Hartogs domain, which is the Hartogs domain over the product of representative bounded homogeneous domains and compute the explicit form of the Bergman kernel function. As an application of this formula, we show that the generalized Bergman–Hartogs domain is a minimal domain with the origin as center. In particular, we get the volume of this domain. In addition, we also study the Lu Qi-Keng problem of the generalized Bergman–Hartogs domain.