Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 537–549

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



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.


Lu Qi-Keng problem Bergman kernel Routh-Hurwitz theorem Jacobi polynomial 

MSC 2010

32A25 33D70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Beberok: Lu Qi-Kengs problem for intersection of two complex ellipsoids. Complex Anal. Oper. Theory 10 (2016), 943–951.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    S. Bergman: Zur Theorie von pseudokonformen Abbildungen. Mat. Sb. (N. S.) 1 (1936), 79–96.MATHGoogle Scholar
  3. [3]
    H. P. Boas, S. Fu, E. J. Straube: The Bergman kernel function: Explicit formulas and zeroes. Proc. Amer. Math. Soc. 127 (1999), 805–811.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J. P. D’Angelo: An explicit computation of the Bergman kernel function. J. Geom. Anal. 4 (1994), 23–34.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi: Higher Transcendental Functions. Vol. 1, Bateman Manuscript Project, McGraw-Hill, New York, 1953.MATHGoogle Scholar
  6. [6]
    S. G. Krantz: Geometric Analysis of the Bergman Kernel and Metric. Graduate Texts in Mathematics 268, Springer, New York, 2013.Google Scholar
  7. [7]
    D. D. Šiljak, D. M. Stipanović: Stability of interval two-variable polynomials and quasipolynomials via positivity. Positive Polynomials in Control (D. Henrion, et al., eds.). Lecture Notes in Control and Inform. Sci. 312, Springer, Berlin, 2005, pp. 165–177.Google Scholar
  8. [8]
    X. Wang: Recursion formulas for Appell functions. Integral Transforms Spec. Funct. 23 (2012), 421–433.MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations