## An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC. (arXiv:1809.04948v1 [math.CA])

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 neigh 查看全文>>