The Bergman kernel function for intersections of some cylindrical domains and Lauricella's hypergeometric function

Abstract

In this paper, we show that Lauricella's hypergeometric function F8 has a close connection with the Bergman kernel for the intersection of two cylindrical domains defined byD(p1,p2,p3):={z∈C3:|z1|2p1+|z2|2p2<1,|z1|2p1+|z3|2p3<1}. We investigate the boundary behavior of the Bergman kernel on the diagonal (z1,0,0). We also compute the explicit form of the Bergman kernel when (p1,p2,p3)=(1,p2,p3) and (p,1,1). As a consequence, we show that D(1,p2,p3) is a Lu Qi-Keng domain. All results can be generalized to the intersection of cylindrical domains in any higher dimension.