An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Let {phi(i)}(i=0)(infinity) be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure mu, that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say E-n(mu), of random polynomials P-n(z):=Sigma(n)(i=0) eta i phi i(z), where eta(0), ...,eta(n) are i.i.d. standard Gaussian random variables. When mu is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that E-n(vertical bar d xi vertical bar) admits an asymptotic expansion of the form E-n(vertical bar d xi vertical bar) similar to 2/pi log(n+1) + Sigma(infinity)(p=0)A(p)(n+1)(-p) (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where mu, is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case E-n(mu) admits an analogous expansion with the coefficients A(p), depending on the measure mu for p >= 1 (the leading order term and A(0) remain the same). (C) 2018 Elsevier Inc. All rights reserved.