Kissing polynomials: algebraic and asymptotic properties

We consider orthogonal polynomials $p_n(x)$ on $[-1,1]$ with respect to the weight $\exp(iwx)$, where $w$ is a real parameter, possibly large. These orthogonal polynomials are relevant in the construction of optimal complex Gaussian quadrature rules, and because of the complex nature of the weight function, they and their Hankel determinants exhibit interesting behaviours, both for finite $n$ and $w$ as well as in the different asymptotic limits. We will present several results in this direction, using both Riemann-Hilbert methods and combinatorics of highly oscillatory multivariate integrals. Based on joint work with Andrew F. Celsus (University of Cambridge, UK), Daan Huybrechs (KU Leuven, Belgium) and Arieh Iserles (University of Cambridge, UK).