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In this paper, we investigate algebraic, differential and asymptotic properties of polynomials p n(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.
orthogonal polynomials; asymptotic approximation in the complex domain; numerical analysis; hankel determinants