On the existence of short trajectories of quadratic differentials related to generalized Jacobi polynomials with non real varying parameters

TL;DR

This paper investigates the conditions under which certain quadratic differentials, related to generalized Jacobi polynomials with non-real parameters, possess finite critical trajectories, impacting the understanding of zero distributions.

Contribution

It provides a necessary and sufficient condition for the existence of finite critical trajectories of a specific quadratic differential associated with generalized Jacobi polynomials.

Findings

Characterization of when finite critical trajectories exist

Connection between quadratic differentials and zero distributions

Conditions involving complex parameters a, b, and λ

Abstract

The study of the asymptotic distributions of zeros of generalized Jacobi polynomials with non real varying parameters, leads with quadratic differentials. In fact, the support of the limit measure of the root-counting measures sits on the finite critical trajectories of a related quadratic differential. In this paper, we study the trajectories of this quadratic differential, more precisely, we give a necessary and sufficient condition on the complex numbers a; b; and {\lambda} for the existence of at list one finite critical trajectory of the quadratic differential (({\lambda}^2(z-a)(z-b))/((z^2-1)^2))dz^2.