Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials

TL;DR

This paper studies multi-orthogonal polynomials related to Nikishin systems on star-like sets, establishing algebraic properties, recurrence relations, zero distribution, and asymptotic behavior under regularity conditions.

Contribution

It proves the normality of the Nikishin system, derives a specific recurrence relation, and describes the asymptotic zero distribution and polynomial behavior.

Findings

Nikishin system is normal with simple roots in the star-like set

Polynomials satisfy a specific (p+1)-term recurrence relation

Asymptotic zero distribution described by a vector equilibrium problem

Abstract

Polynomials $Q_n(z)$, $n=0,1,\ldots,$ that are multi-orthogonal with respect to a Nikishin system of $p\geq 1$ compactly supported measures over the star-like set of $p+1$ rays $S_+:=\{z\in \mathbb{C}: z^{p+1}\geq 0 \}$ are investigated. We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order $p+1$ of the form $z Q_{n}(z)=Q_{n+1}(z)+a_{n}\,Q_{n-p}(z)$ with $a_n>0$ for all $n\geq p$, and that the nonzero roots of $Q_n$ are all simple and located in $S_+$. Under the assumption of regularity (in the sense of Stahl and Totik) of the measures generating the Nikishin system, we describe the asymptotic zero distribution and weak behavior of the polynomials $Q_n$ in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, a theorem on the convergence of the Hermite-Pad\'e approximants to the Nikishin system of Cauchy transforms is proven.