Asymptotics of orthogonal polynomials and the Painlev\'e transcendents
Dan Dai
TL;DR
This survey reviews asymptotic behaviors of orthogonal polynomials linked to Painlevé transcendents, using Riemann-Hilbert problem techniques, and discusses open problems in the field.
Contribution
It provides a comprehensive overview of asymptotic results for orthogonal polynomials related to Painlevé equations using the Deift-Zhou method.
Findings
Asymptotic descriptions of orthogonal polynomials are connected to Painlevé transcendents.
Application of Deift-Zhou nonlinear steepest descent method to Riemann-Hilbert problems.
Identification of open problems in the asymptotic analysis of orthogonal polynomials.
Abstract
In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlev\'e transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the last part of this article, we list several open problems.
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Taxonomy
TopicsMathematical functions and polynomials · Iterative Methods for Nonlinear Equations · Fractional Differential Equations Solutions