Abstract
AbstractWe study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, , over the interval , where is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as have recently been studied for , and our main goal is to extend these results to all 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 is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter approaches a breaking curve, by considering double scaling limits as 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 or some other points on the breaking curve.
Highlights
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
In order to address this apparent schism between the two regimes, the authors of [9] proposed a new quadrature rule based on monic polynomials which satisfy pωn(z)zkeiωz dz = 0
Following the general theory presented in [16], the Hankel determinant of the corresponding family of orthogonal polynomials is closely related to isomonodromic deformations of a certain system of ODEs; more precisely, we consider the vector pn(z; s) = [pn(z, s), pn−1(z; s)]T, that satisfies both a linear system of ODEs in the variable z, as well as an auxiliary linear system of ODEs in the parameter s; compatibility between these two systems of ODEs characterizes the isomodromic deformations of the differential system in z, see [41] and [35, Chapter 4]. Both linear systems can be obtained by standard techniques from the Riemann–Hilbert problem for the OPs, that we present below, and they can be checked to coincide with the linear system corresponding to the Painleve V equation, as given by Jimbo and Miwa in [42], with suitable changes of variable to locate the Fuchsian singularities at z = 0, 1, ∞
Summary
We discuss the necessary background on non-Hermitian orthogonality and state our main findings. We will see that h-functions corresponding to Riemann surfaces of different genus lead to asymptotic expansions which possess markedly different behavior as n → ∞ This difference is analogous to the difference in asymptotic behavior of the polynomials (and their recurrence coefficients) in the one cut and two cut cases, as described above for the GRS program. In order to study the asymptotics of the recurrence coefficients as s → 2, we take s in a double scaling regime near this critical point as s = 2 + L2 , n2/3 where we impose that L2 < 0 This leads us to our final main finding. We end the paper with a proof of Theorem 2.5
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