An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials
Let $ \{φ_i(z;α)\}_{i=0}^\infty $, corresponding to $ α\in(-1,1) $, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(α) $, of random polynomials \[ P_n(z) := \sum_{i=0}^nη_iφ_i(z;α), \] where $ η_0,\dots,η_n $ are i.i.d. standard Gaussian random variables. When $ α=0 $, $ φ_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(α) $ for $ α\neq 0 $. As it turns out, the leading term of the asymptotics in this case is $ (1/π)\log(n+1) $.