Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature

TL;DR

This paper investigates the asymptotic distribution of complex orthogonal polynomials related to Gaussian quadrature, revealing zero accumulation along specific contours with the S-property and deriving strong asymptotics via Riemann-Hilbert analysis.

Contribution

It provides a detailed analysis of the zero distribution and asymptotics of complex orthogonal polynomials associated with Gaussian quadrature on complex contours.

Findings

Zeros accumulate along contours with the S-property.

Strong asymptotics obtained via Deift--Zhou steepest descent method.

Limit distribution of zeros characterized in the complex plane.

Abstract

In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral on the real axis with a high order stationary point, and their limit distribution is also analyzed. We show that the zeros accumulate along a contour in the complex plane that has the S-property in an external field. In addition, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann--Hilbert problem.