The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials

summary:We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2) \in \mathbb {C}^{n+2} \colon |z_1|^2 + \cdots + |z_n|^2 + |w_1|^q < 1, \ |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}.$ We also compute the kernel function for $\{(z_1,w_1,w_2) \in \mathbb {C}^3 \colon |z_1|^{2/n} + |w_1|^q < 1, \ |z_1|^{2/n} + |w_2|^r < 1\}$ and show deflation type identity between these two domains. Moreover in the case that $q=r=2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem