2020/08/19 by Ahmad Barhoumi, Barhoumi, Ahmad, Andrew F. Celsus +3
Mathematics · #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2008.08724
openalex publication_date 2020/08/19 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study a family of monic orthogonal polynomials which are orthogonal with\nrespect to the varying, complex valued weight function, \exp(nsz), over the\ninterval [-1,1], where s\∈\ℂ is arbitrary. This family of\npolynomials originally appeared in the literature when the parameter was purely\nimaginary, that is s\∈ i \ℝ, due to its connection with complex\nGaussian quadrature rules for highly oscillatory integrals. The asymptotics for\nthese polynomials as n\→\∞ have been recently studied for s\∈\ni\ℝ, and our main goal is to extend these results to all s in the\ncomplex plane.\n We first use the technique of continuation in parameter space, developed in\nthe context of the theory of integrable systems, to extend previous results on\nthe so-called modified external field from the imaginary axis to the complex\nplane minus a set of critical curves, called breaking curves. We then apply the\npowerful method of nonlinear steepest descent for oscillatory Riemann-Hilbert\nproblems developed by Deift and Zhou in the 1990s to obtain asymptotics of the\nrecurrence coefficients of these polynomials when the parameter s is away\nfrom the breaking curves. We then provide the analysis of the recurrence\ncoefficients when the parameter s approaches a breaking curve, by considering\ndouble scaling limits as s approaches these points. We shall see a\nqualitative difference in the behavior of the recurrence coefficients,\ndepending on whether or not we are approaching the points s=\± 2 or some\nother points on the breaking curve.\n