The explicit forms and zeros of the Bergman kernel for 3-dimensional Hartogs triangles

We compute the Bergman kernel forHp1,p2,p3={(z1,z2,z3):|z1|p1<|z2|p2<|z3|p3<1}, for any positive integers k=p1,p2,p3 using the transformation formula under a proper holomorphic mapping. For example, we compute the explicit forms of the Bergman kernels of Hk,k,1, H1,k,1, Hk,1,1. Using the explicit formulas, we prove that the Bergman kernels have zeros.