Bidiagonal factorization of tetradiagonal matrices and Darboux transformations

TL;DR

This paper explores the bidiagonal factorization of tetradiagonal matrices, linking it to multiple orthogonal polynomials and Darboux transformations, providing new formulas and insights into their spectral properties.

Contribution

It introduces a novel bidiagonal factorization approach for tetradiagonal matrices and connects it with Darboux transformations and orthogonal polynomial theory.

Findings

Bidiagonal factorization is applicable to certain tetradiagonal matrices.

Darboux transformations are characterized for these matrices.

Christoffel formulas are derived for the factorization elements.

Abstract

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima-Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi-Pi\~neiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin.