[**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.]




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In [arXiv:0310.5297](https://arxiv.org/abs/math/0310297), 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](https://en.wikipedia.org/wiki/Bergman_kernel) is the projection operator from $L^2(\mathbb{D})$ to the Bergman space $A^2(\mathbb{D})$ of $L^2$ holomorphic functions on the unit disk. What is the basis for such a space of functions?



What is the orthonormal basis for the Bergman space on the disk?
Assuming that we normalize so that the area of the unit disc is $\pi$, take $f\_n(z) = \sqrt{\frac{n+1}{\pi}} z^n$ for $n=0,1,2,\dots$ to get an orthonormal basis.


