The kissing polynomials and their Hankel determinants

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 \(w(x)=e^{i\omega x}\) on the interval \([-1,1]\), where \(\omega>0\). We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials \(p_{2n}(x)\) for all values of \(\omega>0\), as well as degeneracy of \(p_{2n+1}(x)\) at certain values of \(\omega\) (called kissing points). We obtain detailed asymptotic information as \(\omega\to\infty\), 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 \(\omega\) asymptotics obtained before.