We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function e niλz on [−1, 1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λ c . Our main goal is to compute their asymptotics when λ > λ c .We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann-Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift-Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane.Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type.In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of λ for which one cannot solve the model Riemann-Hilbert problem, and as such the corresponding polynomials fail to exist. Contents

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.

We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp(nsz), over the interval [−1,1], where s∈C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s∈iR, due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n→∞ have recently been studied for s∈iR, and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s=±2 or some other points on the breaking curve.

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