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 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 $\omega \in \mathbb {R}$, 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.

Global Phase Portrait and Large Degree Asymptotics for the Kissing Polynomials

“…Figures 1, 2, 4, and 5 have been computed using the nonlinear discrete string equations for the recurrence coefficients presented in [11,Theorem 2,Theorem 4], see also [58, §5.2]. In Figure 5, we have used from [21] that β n (s) ∈ R and α n (s) ∈ iR when s ∈ iR. Moreover, it was also shown in [21] that for fixed n, β 2n+1 (it) will have poles (as a function of t) for t ∈ R. As such, we have plotted β 2n+1 on a log scale in Figure 5d.…”

“…where the entries of the matrices m 1 (w) and m 2 (w) are given explicitly in formula (21) in [60], see also [33, (5.0.7)], again in terms of u, u and D (we omit the dependence on w for brevity):…”

“…In the present day, complex orthogonal polynomials with respect to exponential weights have been studied in [15,16] (with quartic potential) and [18,19] (with cubic potential). They have found uses in various areas of mathematics including random matrix theory and theoretical physics [2,3,4,12], rational solutions of Painlevé equations [9,13,14,16], and, of particular interest in the present work, numerical analysis [8,21,26]. Indeed, motivation for the present work is concerned with the numerical treatment of highly oscillatory integrals of the form…”

“…The results of [8] kick started the study of the polynomials in (1.2), and the authors of [21] dubbed such polynomials the Kissing Polynomials on account of the behavior of their zero trajectories in the complex plane. In particular, the work [21] provides the existence of the even degree Kissing polynomials, along with the asymptotic behavior of the polynomials as ω → ∞ with n fixed.…”

“…The results of [8] kick started the study of the polynomials in (1.2), and the authors of [21] dubbed such polynomials the Kissing Polynomials on account of the behavior of their zero trajectories in the complex plane. In particular, the work [21] provides the existence of the even degree Kissing polynomials, along with the asymptotic behavior of the polynomials as ω → ∞ with n fixed. On the other hand, the asymptotic analysis of the Kissing polynomials for fixed ω as n → ∞ can be handled via the Riemann-Hilbert techniques discussed in [41] or the appendix of [25], where it was shown that the zeros of the Kissing polynomials accumulate on the interval [−1, 1] as n → ∞ with ω > 0 fixed.…”