Boundary behavior of the Bergman kernel for generalized Fock-Bargmann-Hartogs domains

Abstract

In this paper, we study the Bergman kernel for the Hartogs type domainDμ,p:={(z,ζ)∈C×Cn:‖ζ‖2<e−μ|z|p}. In particular, we compute the explicit form of the Bergman kernel for Dμ,2m for any positive integer m. The relations between the Mittag-Leffler function and the generalized Fock kernel are investigated. Using the explicit formula, we study the asymptotic behavior of the Fock kernel and the boundary behavior of the Bergman kernel on the diagonal for the generalized Fock-Bargmann-Hartogs domains Dμ,2m.