An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials

Abstract

Let $\{{\phi }_i(z;\alpha ){\}}_{i=0}^{\infty }$, corresponding to $\alpha \in (-1,1)$, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{E}}_n(\alpha )$, of random polynomials \[ P_n(z) := \sum_{i=0}^nη_iφ_i(z;α), \] where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When $\alpha =0$, ${\phi }_i(z;0)=z^i$ and $P_n(z)$ are called Kac polynomials. In this case it was shown by Wilkins that ${\mathbb{E}}_n(0)$ admits an asymptotic expansion of the form \[ \mathbb E_n(0) \sim \frac2π\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $\mathbb{E}(\alpha )$ for $\alpha \ne 0$. As it turns out, the leading term of the asymptotics in this case is $(1/\pi )\log (n+1)$.