  We study a family of polynomials which are orthogonal with respect to the
varying, highly oscillatory complex weight function $e^{ni\lambda z}$ on
$[-1,1]$, where $\lambda$ 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 $\lambda$ is
smaller than a certain critical value, $\lambda_c$. Our main goal is to compute
their asymptotics when $\lambda>\lambda_c$.
  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 $\lambda$ for which one cannot solve the model Riemann-Hilbert
problem, and as such the corresponding polynomials fail to exist.

Classical Analysis and ODEs;Complex Variables