On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials

TL;DR

This paper studies multiple orthogonal polynomials related to Nikishin systems with one measure discrete and unbounded supports, solving an equilibrium problem to understand their zero distribution behavior.

Contribution

It introduces a Nikishin type equilibrium problem with external field and constraints, providing new insights into the asymptotic zero distribution of these polynomials.

Findings

Derived the zero distribution of multiple orthogonal polynomials

Solved a new equilibrium problem with external field and constraints

Extended understanding of Nikishin systems with discrete components

Abstract

We consider multiple orthogonal polynomials with respect to Nikishin systems generated by two measures $(\sigma_1, \sigma_2)$ with unbounded supports ($\mbox{supp} \, \sigma_1 \subseteq \mathbb{R}_+$, $\mbox{supp} \, \sigma_2 \subseteq \,\mathbb{R}_-$) and $\sigma_2$ is discrete. A Nikishin type equilibrium problem in the presence of an external field acting on $\mathbb{R}_+$ and a constraint on $ \mathbb{R}_-$ is stated and solved. The solution is used for deriving the contracted zero distribution of the associated multiple orthogonal polynomials.