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

[1] T. Beberok: Lu Qi-Keng’s problem for intersection of two complex ellipsoids. Complex Anal. Oper. Theory 10 (2016), 943-951. DOI 10.1007/s11785-015-0505-4 | MR 3506300 | Zbl 06601600
[2] S. Bergman: Zur Theorie von pseudokonformen Abbildungen. Mat. Sb. (N.S.) 1 (1936), 79-96. Zbl 0014.02603
[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. DOI 10.1090/S0002-9939-99-04570-0 | MR 1469401 | Zbl 0919.32013
[4] J. P. D'Angelo: An explicit computation of the Bergman kernel function. J. Geom. Anal. 4 (1994), 23-34. DOI 10.1007/BF02921591 | MR 1274136 | Zbl 0794.32021
[5] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi: Higher Transcendental Functions. Vol. 1, Bateman Manuscript Project, McGraw-Hill, New York (1953). MR 0698779 | Zbl 0051.30303
[6] S. G. Krantz: Geometric Analysis of the Bergman Kernel and Metric. Graduate Texts in Mathematics 268, Springer, New York (2013). DOI 10.1007/978-1-4614-7924-6 | MR 3114665 | Zbl 1281.32004
[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. Lecture Notes in Control and Inform. Sci. 312, Springer, Berlin (2005), 165-177. DOI 10.1007/10997703_10 | MR 2123523 | Zbl 1138.93392
[8] X. Wang: Recursion formulas for Appell functions. Integral Transforms Spec. Funct. 23 (2012), 421-433. DOI 10.1080/10652469.2011.596483 | MR 2929185 | Zbl 1273.33013

Affiliations:   Tomasz Beberok, Sabanci University, Orta Mahalle, Universite Caddesi No: 27, Lojmanlari G7-102, Tuzla, 34956 Istanbul, Turkey, e-mail:

Springer subscribers can access the papers on Springer website.
Access to full texts on this site is restricted to subscribers of Myris Trade. To activate your access, please send an e-mail to
[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at: