I am looking for the Bergman kernel of the upper half-space $D={z|\Im{z} > 0, z \in \mathbb{C}^n}$. I know how to derive the kernel for upper half-plane using Riemann mapping theorem, but similar theorem does not exist in several complex variable case.
It is easy to find a biholomorphic map between the poly-disc and upper half-space by individual Mobius transformation of each dimension, would that be a sound way of constructing the Bergman kernel for the upper half-space?