The multiple Markov theorem on Angelesco sets

TL;DR

The paper generalizes Markov’s monotonicity of zeros from classical orthogonal polynomials to monic multiple orthogonal polynomials on Angelesco sets by establishing a differentiability framework for zeros with respect to a parameter $t$ and introducing a block interaction matrix $\mathbf{A}(t)$. Under per-weight Markov-type conditions and a sufficient $\mathcal{M}$-matrix condition on $\mathbf{A}(t)$, zeros of the MOPRL $P_n(x;t)$ are shown to move monotonically with $t$, with the direction dictated by the partition structure; the classical single-weight case is recovered as $m=1$. The approach relies on differentiating the moments and orthogonality relations, applying the implicit function theorem to track zeros, and showing that the relevant matrices are $\mathcal{Z}$- or $\mathcal{M}$-matrices under explicit monotonicity assumptions on the weight derivatives. This yields a general, unrestricted monotonicity result for zeros on Angelesco sets, with potential implications for related special-function theory and integrable structures.

Abstract

By addressing a long-standing open problem, listed in a highly regarded collection of open questions in the field and described as a "worthwhile research project", this note extends Markov's theorem (Markoff, Math. Ann., 27:177-182, 1886) on the variation of zeros of orthogonal polynomials on the real line to the setting of multiple orthogonal polynomials on Angelesco sets. The analysis reveals that the only distinction from the classical 1886 result lies in establishing sufficient conditions for a given $\mathcal{Z}$-matrix--which, in the Markov case, is the identity matrix--to be an $\mathcal{M}$-matrix. In contrast to most existing studies, which often present highly technical proofs for specific results, this note seeks to provide a simple proof of a general result without imposing restrictions on the weight functions (such as their potential "classical" nature), the number of intervals, or the structure of the partition.

Figures (2)

• Figure 1: The support of $\nu_1$ and $\nu_2$.
• Figure 2: The support of $\nu_1$, $\nu_2$, and $\nu_3$.

Theorems & Definitions (5)

• proof
• proof
• proof
• proof
• proof