The kissing polynomials and their Hankel determinants

Abstract

In this paper, we investigate algebraic, differential and asymptotic properties of polynomials pn(x) that are orthogonal with respect to the complex oscillatory weight w(x)=eiωx on the interval [−1,1]⁠, where ω>0⁠. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials p2n(x) for all values of ω∈R⁠, as well as degeneracy of p2n+1(x) at certain values of ω (called kissing points). We obtain detailed asymptotic information as ω→∞⁠, using recent theory of multivariate highly oscillatory integrals, and we complete the analysis with the study of complex zeros of Hankel determinants, using the large ω asymptotics obtained before.