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In this paper we investigate the Bergman kernel function for intersection oftwo complex ellipsoids $\{(z,w_1,w_2) \in \mathbb{C}^{n+2} : |z_1|^2 + \cdots +|z_n|^2 + |w_1|^q < 1, \quad |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}.$

The Bergman kernel: explicit formulas, deflation, Lu Qi-Keng problem and Jacobi Polynomials