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

Abstract

For a system of two measures supported on a starlike set in the complex plane, we study the asymptotic properties of the 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 .