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arXiv · 1504.07297

The kissing polynomials and their Hankel determinants

Andrew F. Celsus·Alfredo Deaño·Daan Huybrechs·Arieh Iserles
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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.

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Andrew F. Celsus, Alfredo Deaño, Daan Huybrechs, Arieh Iserles. 2015-04-27. The kissing polynomials and their Hankel determinants. https://arxiv.org/abs/1504.07297

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