[**EDIT by YC**: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]

In arXiv:0310.5297, Yuval Peres and Balin Virag study the roots of random power series, $f(z) = \sum a_n z^n$ (where the $a_n$ are Gaussian with mean 0 and variance 1) and show that correlations of the roots are determined by the Bergman kernel $$ \rho(z_1, \dots, z_n) = \frac{1}{\pi^n} \det\left[ \frac{1}{1 - z_i \overline{z_j}} \right]_{i,j}$$ According to Wikipedia, the Bergman kernel is the projection operator from $L^2(\mathbb{D})$ to the Hardy space $H^2(\mathbb{D})$ of $L^2$ holomorphic functions on the unit disk. What is the basis for such a space of functions?