We derive two identities for multiple basic hyper-geometric series associated with the unitary $U(n+1)$ group. In order to get the two identities, we first present two known $q$-exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two $U(n+1)$$q$-Chu-Vandermonde summations established by Milne, we arrive at our...

We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2)\in {ℂ}^{n+2}:|z_1{|}^2+\cdots +|z_n{|}^2+|w_1{|}^q<1,|z_1{|}^2+\cdots +|z_n{|}^2+|w_2{|}^r<1\}.$ We also compute the kernel function for $\{(z_1,w_1,w_2)\in {ℂ}^3:|z_1{|}^{2/n}+|w_1{|}^q<1,|z_1{|}^{2/n}+|w_2{|}^r<1\}$ 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.