The kissing polynomials and their Hankel determinants

TL;DR

This paper studies orthogonal polynomials with complex oscillatory weights, analyzing their algebraic, differential, and asymptotic properties, including Hankel determinants and zeros, especially as the oscillation parameter grows large.

Contribution

It establishes existence and degeneracy conditions for these polynomials and provides detailed asymptotic analysis for large oscillation parameters.

Findings

Existence of even-degree polynomials for all >0

Degeneracy of odd-degree polynomials at specific  values

Asymptotic behavior of polynomials and Hankel determinants as 

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 $w(x)=e^{i\omega x}$ 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.