Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 537-549, 2017


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

Tomasz Beberok

Received February 15, 2016.  First published March 1, 2017.

Abstract:  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.
Keywords:  Lu Qi-Keng problem; Bergman kernel; Routh-Hurwitz theorem; Jacobi polynomial
Classification MSC:  32A25, 33D70
DOI:  10.21136/CMJ.2017.0073-16


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Affiliations:   Tomasz Beberok, Sabanci University, Orta Mahalle, Universite Caddesi No: 27, Lojmanlari G7-102, Tuzla, 34956 Istanbul, Turkey, e-mail: tbeberok@ar.krakow.pl

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