English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We study monic polynomials $Q_n(x)$ generated by a high order three-termrecursion $xQ_n(x)=Q_{n+1}(x)+a_{n-p} Q_{n-p}(x)$ with arbitrary $p\geq 1$ and$a_n>0$ for all $n$. The recursion is encoded by a two-diagonal Hessenbergoperator $H$. One of our main results is that, for periodic coefficients $a_n$and under certain conditions, the $Q_n$ are multiple orthogonal polynomialswith respect to a Nikishin system of orthogonality measures supported onstar-like sets in the complex plane. This improves a recent result ofAptekarev-Kalyagin-Saff where a formal connection with Nikishin systems wasobtained 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", orequivalently, the "generalized eigenvalues" of the Hessenberg matrix $H$. Weprove interlacing relations for the generalized eigenvalues by using totallypositive matrices. In the case of asymptotically periodic coefficients $a_n$,we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtaina connection with a vector equilibrium problem. We anticipate that in thefuture, the study of Riemann-Hilbert minors may prove useful for more generalclasses of multiple orthogonal polynomials.

High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets