Banded totally positive matrices and normality for mixed multiple orthogonal polynomials

TL;DR

This paper explores banded totally positive matrices, demonstrating their semigroup structure and positive bidiagonal factorizations, and applies these concepts to establish normality and spectral properties of mixed multiple orthogonal polynomials.

Contribution

It introduces new characterizations of banded totally positive matrices and links their structure to the normality of mixed multiple orthogonal polynomials on the step line.

Findings

Banded totally positive matrices form a semigroup.

Existence of positive bidiagonal factorizations within these matrices.

Spectral Favard theorem guarantees measures for mixed multiple orthogonal polynomials.

Abstract

This paper serves as an introduction to banded totally positive matrices, exploring various characterizations and associated properties. A significant result within is the demonstration that the collection of such matrices forms a semigroup, notably including a subset permitting positive bidiagonal factorization. Moreover, the paper applies this concept to investigate step line normality concerning the degrees of associated recursion polynomials. It presents a spectral Favard theorem, ensuring the existence of measures, thereby guaranteeing that these recursion polynomials represent mixed multiple orthogonal polynomials that maintain normality on the step line indices.