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

Abstract

We investigate the ratio asymptotic behavior of the sequence (Qn)n=0∞ 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∈C:zp+1≥0}. The main algebraic property of these polynomials is that they satisfy a three-term recurrence relation of the form zQn(z)=Qn+1(z)+anQn−p(z) with an>0 for all n≥p. Under a Rakhmanov-type condition on the measures generating the Nikishin system, we prove that the sequence of ratios Qn+1(z)∕Qn(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 Miña-Díaz 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.