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

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 $p\ge 1$ measures that are compactly supported on the star-like set of $p+1$ rays $S_{+}=\{z\in \mathbb{C}:z^{p+1}\ge 0\}$. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form $z{Q}_n(z)=Q_{n+1}(z)+a_n{Q}_{n-p}(z)$ with $a_n>0$ for all $n\ge p$. 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 Miña-Díaz 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 $p=2$.

arXiv commentMinor corrections were done for this version. A continuation of this work is in arXiv: 1907.03002. This paper has 37 pages, 1 table, no figures