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

Abstract

Let {φi(z;α)}i=0∞, corresponding to α∈(−1,1), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say 𝔼n(α), of random polynomials Pn(z):= ∑i=0nηiφi(z;α), where η0,…,ηn are i.i.d. standard Gaussian random variables. When α=0, φi(z;0)=zi and Pn(z) are called Kac polynomials. In this case it was shown by Wilkins that 𝔼n(0) admits an asymptotic expansion of the form 𝔼n(0)∼2πlog(n+1)+ ∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of 𝔼(α) for α≠0. As it turns out, the leading term of the asymptotics in this case is (1∕π)log(n+1).