Zeros of new Bergman kernels

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.

6 figures, Comments are welcome