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We study monic polynomials  () generated by a high-order three-term recursion  ()= ()+   () with arbitrary ≥1 and  >0 for all . The recursion is encoded by a two-diagonal Hessenberg operator . One of our main results is that, for periodic coefficients   and under certain conditions, the   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  for some >0.An important tool in this paper is the study of ‘Riemann–Hilbert minors’, or equivalently, the ‘generalized eigenvalues’ of the Hessenberg matrix . We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients  , 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.

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