calculating Bergman kernel of punctured unit disk

Question

Any idea how can we calculate the Bergman kernel of the punctured unit disk? I know how to find the Bergman kernel of the unit disk, for example, we have complete orthonormal basis for the Bergman space $A^2(\mathbb{D})$.

Is it true that the complete orthonormal basis of $A^2(\mathbb{D}^{*})$ same as $A^2(\mathbb{D})$?

Answer 1

I proved that the two spaces have the same complete orthonormal basis, $\sqrt{\dfrac{n+1}{\pi}}z^n$, $n=0,1,2, \cdots$. Thus both spaces have have the same kernel.