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

作者：Garcia A Lopez^*

来源：Journal of Approximation Theory, 2011, 163(9): 1146-1184.

DOI：10.1016/j.jat.2011.03.008

• Higher-order three-term recurrences
• Nikishin systems
• Ratio asymptotics
• nth root asymptotics
• Zero asymptotic distribution

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 nth root asymptotic behavior and zero asymptotic distribution of Q(n).

• 出版日期2011-9