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

We investigate the ratio asymptotic behavior of the sequence (Q(n))(n=0)(infinity) of multiple orthogonal polynomials associated with a Nikishin system of p >= 1 measures that are compactly supported on the star-like set of p + 1 rays S+ = {z is an element of C : Z(P+1) >= 0}. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form zQ(n)(Z) = Q(n+1)(Z) + a(n) Q(n-p) (z) with a(n) > 0 for all n >= 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 an of recurrence coefficients are limit periodic with period p(p + 1). Our results complement some results obtained by the first author and Mifia-Diaz 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. (C) 2017 Elsevier Inc. All rights reserved.