The kissing polynomials and their Hankel determinants

We study a family of polynomials that are orthogonal with respect to the weight function e in [−1, 1], where ω ≥ 0. Since this weight function is complex-valued and, for large ω, highly oscillatory, many results in the classical theory of orthogonal polynomials do not apply. In particular, the polynomials need not exist for all values of the parameter ω, and, once they do, their roots lie in the complex plane. Our results are based on analysing the Hankel determinants of these polynomials, reformulated in terms of high-dimensional oscillatory integrals which are amenable to asymptotic analysis. This analysis yields existence of the even-degree polynomials for large values of ω, an asymptotic expansion of the polynomials in terms of rescaled Laguerre polynomials near ±1 and a description of the intricate structure of the roots of the Hankel determinants in the complex plane. This work is motivated by the design of efficient quadrature schemes for highly oscillatory integrals.