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

Abstract

Let ${\left\{{\phi }_i\left(z;\alpha \right)\right\}}_{i=0}^{\infty }$, corresponding to $\alpha \in \left(-1,1\right)$, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{𝔼}}_n\left(\alpha \right)$, of random polynomials

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

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When $\alpha =0$, ${\phi }_i\left(z;0\right)=z^i$ and $P_n\left(z\right)$ are called Kac polynomials. In this case it was shown by Wilkins that ${\mathbb{𝔼}}_n\left(0\right)$ admits an asymptotic expansion of the form

$${\mathbb{𝔼}}_n\left(0\right)\sim \frac{2}{\pi }log\left(n+1\right)+\sum _{p=0}^{\infty }{A}_p{\left(n+1\right)}^{-p}$$

(Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $\mathbb{𝔼}\left(\alpha \right)$ for $\alpha \ne 0$. As it turns out, the leading term of the asymptotics in this case is $\left(1∕\pi \right)log\left(n+1\right)$.