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

Let ${\{{\phi }_i\}}_{i=0}^{\infty }$ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ 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

$P_n(z):=\sum _{i=0}^n{\eta }_i{\phi }_i(z),$

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When μ 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

${\mathbb{E}}_n(|\mathrm{d}\xi |)～\frac{2}{\pi }\log ?(n+1)+\sum _{p=0}^{\infty }{A}_p{(n+1)}^{?p}$

(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ 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 the coefficients $A_p$ depending on the measure μ for $p\ge 1$ (the leading order term and $A_0$ remain the same).