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

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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 є C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s є 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 є 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.
Asymptotic analysis, Continuation in parameter space, Orthogonal polynomials in the complex plane, Riemann-Hilbert problem
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Barhoumi, A., Celsus, A. F. & Deaño, A. (2021). Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials. Studies in Applied Mathematics, 147(2), pp. 417-833.