Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

TL;DR

This paper generalizes the connection between orthogonal polynomials and Jacobi matrices to multiple orthogonal polynomials on trees, providing new insights and asymptotic results through operator theory.

Contribution

It introduces self-adjoint Jacobi matrices on rooted trees associated with multiple orthogonal polynomials, extending classical one-dimensional results to higher dimensions.

Findings

Expressed Green's functions in terms of MOPs

Proved ratio asymptotics for MOPs

Connected polynomial theory with operator theory on trees

Abstract

We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.