High-Order Three-Term Recursions, Riemann–Hilbert Minors and Nikishin Systems on Star-Like Sets

Abstract

We study monic polynomials Q n (x) generated by a high-order three-term recursion xQ n (x)=Q n+1(x)+a n−p Q n−p (x) with arbitrary p≥1 and a n >0 for all n. The recursion is encoded by a two-diagonal Hessenberg operator H. One of our main results is that, for periodic coefficients a n and under certain conditions, the Q n are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev–Kalyagin–Saff, where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_{n}-a|<\infty$ for some a>0. An important tool in this paper is the study of ‘Riemann–Hilbert minors’, or equivalently, the ‘generalized eigenvalues’ of the Hessenberg matrix H. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients a n , we find weak and ratio asymptotics for the Riemann–Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann–Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.