Bidiagonal matrix factorisations related to multiple orthogonal polynomials

TL;DR

This paper characterizes when Hessenberg matrices linked to multiple orthogonal polynomials can be factorized into bidiagonal matrices, connecting these factorizations to continued fractions and providing explicit examples for specific polynomial systems.

Contribution

It establishes necessary and sufficient conditions for bidiagonal factorization of Hessenberg matrices associated with multiple orthogonal polynomials and relates these to continued fractions and polynomial coefficients.

Findings

Derived conditions for bidiagonal factorization of Hessenberg matrices.

Connected bidiagonal entries to coefficients of multiple orthogonal polynomials and continued fractions.

Provided explicit factorization for Jacobi-Piñeiro and multiple Laguerre polynomials.

Abstract

We provide necessary and sufficient conditions for the Hessenberg recurrence matrix associated with a system of multiple orthogonal polynomials to admit a factorisation as a product of bidiagonal matrices. Using the Gauss-Borel factorisation of the moment matrix, we show that the nontrivial entries of those bidiagonal matrices can be expressed in terms of coefficients of type I or type II multiple orthogonal polynomials on the step-line with respect to the original system and its Christoffel transformations. Using the connection of multiple orthogonal polynomials with branched continued fractions, we show that the nontrivial entries of the bidiagonal matrices in the factorisation of the Hessenberg recurrence matrix correspond to the coefficients of a branched continued fraction associated with the given system of multiple orthogonal polynomials. As a case study, we present an explicit bidiagonal factorisation for the Hessenberg recurrence matrices of the Jacobi-Pi\~neiro polynomials and, as a limiting case, the multiple Laguerre polynomials of first kind.