We study monic polynomials Qn(x) generated by a high order three-term
recursion xQn(x)=Qn+1(x)+an−pQn−p(x) with arbitrary p≥1 and
an>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 an
and under certain conditions, the Qn 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 ∑n=0∞∣an−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 an,
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.Comment: 59 pages, 3 figure