  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.
