Trajectories of quadratic differentials for Jacobi polynomials with complex parameters

TL;DR

This paper analyzes the trajectories of quadratic differentials related to Jacobi polynomials with complex parameters to understand their zero distribution in the complex plane.

Contribution

It establishes the global structure of trajectories for a specific quadratic differential and describes the asymptotic zero distribution of Jacobi polynomials with complex parameters.

Findings

The zero distribution concentrates on an open arc in the complex plane.

The support of the zero distribution is characterized by a critical trajectory.

The structure of trajectories is linked to symmetry properties of an equilibrium measure.

Abstract

Motivated by the study of the asymptotic behavior of Jacobi polynomials $\left( P_{n}^{(nA,nB)}\right) _{n}$ with $A\in \mathbb C$ and $B>0$ we establish the global structure of trajectories of the related rational quadratic differential on $\mathbb C$. As a consequence, the asymptotic zero distribution (limit of the root-counting measures of $\left( P_{n}^{(nA,nB)}\right) _{n}$) is described. The support of this measure is formed by an open arc in the complex plan (critical trajectory of the aforementioned quadratic differential) that can be characterized by the symmetry property of its equilibrium measure in a certain external field.