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

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

2010-09-02

arxiv.org/abs/1009.0466v1

Mathematics

Complex Variables

31 pages

Scientific paper

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 $Q_{n}$.

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