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

Tomasz Beberok

arXiv:1601.02989·math.CV·January 13, 2016

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

Tomasz Beberok

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TL;DR

This paper derives explicit formulas for the Bergman kernel on complex ellipsoid intersections, explores the Lu Qi-Keng problem, and connects these kernels to Jacobi polynomials, advancing understanding of complex analysis in several variables.

Contribution

It provides new explicit formulas for the Bergman kernel on complex ellipsoid intersections and investigates related problems like the Lu Qi-Keng problem.

Findings

01

Explicit formulas for the Bergman kernel on the intersection of two complex ellipsoids.

02

Analysis of the Lu Qi-Keng problem in this context.

03

Connections established between the Bergman kernel and Jacobi polynomials.

Abstract

In this paper we investigate the Bergman kernel function for intersection of two complex ellipsoids ${(z, w_{1}, w_{2}) \in C^{n + 2} : ∣ z_{1} ∣^{2} + \dots + ∣ z_{n} ∣^{2} + ∣ w_{1} ∣^{q} < 1, ∣ z_{1} ∣^{2} + \dots + ∣ z_{n} ∣^{2} + ∣ w_{2} ∣^{r} < 1} .$

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Taxonomy

TopicsHolomorphic and Operator Theory · Mathematical functions and polynomials · Algebraic and Geometric Analysis