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

TL;DR

This paper derives an asymptotic expansion for the expected number of real zeros of random orthonormal Geronimus polynomials, generalizing known results for Kac polynomials and revealing a logarithmic growth rate.

Contribution

It extends asymptotic analysis of real zeros from Kac polynomials to a broader class of Geronimus polynomials for all th.

Findings

Expected zeros grow logarithmically with degree n.

Leading term of the asymptotic expansion is (1/th) log(n+1).

Provides a unified asymptotic framework for different th.

Abstract

Let $ \{\varphi_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 \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z;\alpha), \] where $ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $ \alpha=0 $, $ \varphi_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 \[ \mathbb E_n(0) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $ \mathbb E(\alpha) $ for $ \alpha\neq 0 $. As it turns out, the leading term of the asymptotics in this case is $ (1/\pi)\log(n+1) $.