Zeros of new Bergman kernels

TL;DR

This paper explicitly determines the kernels of new Bergman spaces on the unit disk and analyzes the distribution of their zeros, especially when parameters are integers, supported by numerical results.

Contribution

It provides explicit formulas for the kernels of new Bergman spaces and studies the zero distribution, including polynomial zero characterization and numerical analysis.

Findings

Zeros are characterized by roots of a real polynomial Q_{α,β}.

Explicit kernel formulas are derived for the new Bergman spaces.

Numerical results illustrate the zero distribution patterns.

Abstract

In this paper we determine explicitly the kernels $\mathbb K_{\alpha,\beta}$ associated with new Bergman spaces $\mathcal A_{\alpha,\beta}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $\alpha\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{\alpha,\beta}$. Some numerical results are given throughout the paper.