  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).

Classical Analysis and ODEs