  In this note, we prove a central limit theorem for smooth linear statistics
of zeros of random polynomials which are linear combinations of orthogonal
polynomials with iid standard complex Gaussian coefficients. Along the way, we
obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials
endowed with varying measures of the form $e^{-2n\varphi_n(z)}dz$ under
suitable assumptions on the weight functions $\varphi_n$.
