Zeros of complex random polynomials spanned by Bergman polynomials

We study the expected number of zeros $$P_n(z)=\sum_{k=0}^n\eta_kp_k(z),$$ where $\{\eta_k\}$ are complex-valued i.i.d standard Gaussian random variables, and $\{p_k(z)\}$ polynomials orthogonal on unit disk. When $p_k(z)=\sqrt{(k+1)/\pi} z^k$, $k\in \{0,1,\dots, n\}$, we give an explicit formula for $P_n(z)$ in a disk radius $r\in (0,1)$ centered at origin. From our establish limiting value zeros, radially expanding disk, show that is $2n/3$. Generalizing basis functions to be regular sense Ullman--Stahl--Totik, measure orthogonality associated absolutely continuous with respect planar Lebesgue measure, origin, asymptotically