Global-phase portrait and large-degree asymptotics for the Kissing polynomials

Abstract

Studies in Applied Mathematics published by Wiley Periodicals LLCWe study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex-valued weight function, (Formula presented.), over the interval (Formula presented.), where (Formula presented.) is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, (Formula presented.), due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as (Formula presented.) have recently been studied for (Formula presented.), and our main goal is to extend these results to all (Formula presented.) 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 (Formula presented.) is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter (Formula presented.) approaches a breaking curve, by considering double scaling limits as (Formula presented.) 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 (Formula presented.) or some other points on the breaking curve.