RT Journal Article
T1 Global-phase portrait and large-degree asymptotics for the Kissing polynomials
A1 Barhoumi, A.
A1 Celsus, A. F.
A1 Deaño Cabrera, Alfredo
AB 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.
PB Wiley
SN 0022-2526
YR 2021
FD 2021-08
LK https://hdl.handle.net/10016/36263
UL https://hdl.handle.net/10016/36263
LA eng
NO This work was carried out while A.F.C. was a PhD student at the University of Cambridge, and he is thankful for his current support by the Cantab Capital Insti-tute for the Mathematics of Information and the Cambridge Centre for Analysis. A. D. gratefully acknowledges financial support from EPSRC, grant EP/P026532/1, Painlevé equations: analytical properties and numerical computation, as well as from the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors(EPUC3M23),and in the context of the VPRICIT (Regional Programme of Research and Technological Innovation).
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RD 1 sept. 2024