RT Journal Article
T1 Global-phase portrait and large-degree asymptotics for the Kissing polynomials
A1 Barhoumi, Ahmad
A1 Celsus, Andrew F.
A1 Deaño Cabrera, Alfredo
AB We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complexvalued weight function, exp (nsz), over the interval [−1, 1], where s∈ℂ is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, 𝑠��∈𝑖��ℝ, 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��∈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 notwe 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/10016/34392
UL https://hdl.handle.net/10016/34392
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 Institute 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(RegionalProgrammeofResearchand Technological Innovation).
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RD 24 may. 2024