The kissing polynomials and their Hankel determinants Articles uri icon

publication date

  • January 2022

start page

  • 1

end page

  • 66


  • 1, tnab005


  • 6

Electronic International Standard Serial Number (EISSN)

  • 2398-4945


  • 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