Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 537–549

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

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Abstract

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w1,w2) ∈ Cn+2: |z1|2+...+|zn|2+|w1|q < 1, |z1|2+...+|zn|2+|w2|r < 1}. We also compute the kernel function for {(z1,w1,w2) ∈ C3: |z1|2/n + |w1|q < 1, |z1|2/n + |w2|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 

MSC 2010

32A25 33D70 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Sabanci University, Orta MahalleIstanbulTurkey

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