The kissing polynomials and their Hankel determinants

Abstract 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 $w(x)=\mathrm {e}^{\mathrm {i}\omega x}$ on interval $[-1,1]$, where $\omega>0$. We also related quantities such as Hankel determinants recurrence coefficients. prove existence $p_{2n}(x)$ for all values $\omega \in \mathbb {R}$, well degeneracy $p_{2n+1}(x)$ at certain $ (called kissing points). obtain detailed information \to \infty $, using recent theory multivariate highly integrals, complete analysis study zeros determinants, large asymptotics obtained before.