Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials

TL;DR

This paper studies the ratio asymptotics of multiple orthogonal polynomials associated with Nikishin systems on star-like sets, revealing their limit periodic behavior under certain conditions.

Contribution

It establishes the limit periodicity of ratio asymptotics and recurrence coefficients for these polynomials, extending previous results to more general star-like sets.

Findings

Ratios of consecutive polynomials are limit periodic with period p(p+1)

Recurrence coefficients a_n are limit periodic with period p(p+1)

Results extend previous work to more general star-like sets

Abstract

We investigate the ratio asymptotic behavior of the sequence $(Q_{n})_{n=0}^{\infty}$ of multiple orthogonal polynomials associated with a Nikishin system of $p\geq 1$ measures that are compactly supported on the star-like set of $p+1$ rays $S_{+}=\{z\in\mathbb{C}: z^{p+1}\geq 0\}$. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form $zQ_{n}(z)=Q_{n+1}(z)+a_{n} Q_{n-p}(z)$ with $a_{n}>0$ for all $n\geq p$. Under a Rakhmanov-type condition on the measures generating the Nikishin system, we prove that the sequence of ratios $Q_{n+1}(z)/Q_{n}(z)$ and the sequence $a_{n}$ of recurrence coefficients are limit periodic with period $p(p+1)$. Our results complement some results obtained by the first author and Mi\~{n}a-D\'{i}az in a recent paper in which algebraic properties and weak asymptotics of these polynomials were investigated. Our results also extend some results obtained by the first author in the case $p=2$.