Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set

TL;DR

This paper investigates the asymptotic behavior of multiple orthogonal polynomials associated with a two-measure system supported on a starlike set, revealing periodic limits and zero distribution patterns.

Contribution

It provides new insights into the asymptotics of multiple orthogonal polynomials for Nikishin-type systems on starlike sets, including limit behaviors and relations to Riemann surface conformal mappings.

Findings

Ratios of consecutive polynomials have four distinct periodic limits.

Limit functions are described via conformal mappings of Riemann surfaces.

Zero distribution of polynomials follows specific asymptotic patterns.

Abstract

For a system of two measures supported on a starlike set in the complex plane, we study asymptotic properties of associated multiple orthogonal polynomials $Q_{n}$ and their recurrence coefficients. These measures are assumed to form a Nikishin-type system, and the polynomials $Q_{n}$ satisfy a three-term recurrence relation of order three with positive coefficients. Under certain assumptions on the orthogonality measures, we prove that the sequence of ratios $\{Q_{n+1}/Q_{n}\}$ has four different periodic limits, and we describe these limits in terms of a conformal representation of a compact Riemann surface. Several relations are found involving these limiting functions and the limiting values of the recurrence coefficients. We also study the $n$th root asymptotic behavior and zero asymptotic distribution of $Q_{n}$.