An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Abstract: Let $ {\varphi_i}{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure $ \mu $ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(\mu) $, of random polynomials [ P_n(z) := \sum{i=0}^n\eta_i\varphi_i(z), ] where $ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $ \mu $ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that $ \mathbb E_n(|\mathrm d\xi|) $ admits an asymptotic expansion of the form [ \mathbb E_n(|\mathrm d\xi|) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} ] (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where $ \mu $ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case $ \mathbb E_n(\mu) $ admits an analogous expansion with coefficients the $ A_p $ depending on the measure $ \mu $ for $ p\geq 1 $ (the leading order term and $ A_0 $ remain the same).