Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

We consider polynomials pn(x) that are orthogonal with respect to the oscillatory weight w(x) = e on [−1, 1], where ω > 0 is a real parameter. A first analysis of pn(x) for large values of ω was carried out in [5], in connection with complex Gaussian quadrature rules with uniform good properties in ω. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of pn(x) in the complex plane as n→∞. The parameter ω grows with n linearly. The tools used are logarithmic potential theory and the S-property, together with the Riemann–Hilbert formulation and the Deift–Zhou steepest descent method.