Bidiagonal matrix factorisations related to multiple orthogonal polynomials

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-Piñeiro polynomials and, as a limiting case, the multiple Laguerre polynomials of first kind.

💡 Research Summary

This research paper presents a rigorous mathematical investigation into the bidiagonal factorization of the Hessenberg recurrence matrix associated with a system of multiple orthogonal polynomials (MOPs). The central problem addressed is determining the necessary and sufficient conditions under which the $(r+1)$-banded Hessenberg matrix $H$, which arises from $r$ linear functionals, can be decomposed into a product of $un$ lower-bidiagonal matrices ($L_1, \dots, L_r$) and a single upper-bidiagonal matrix ($U$).

The authors employ the Gauss-Borel factorization of the moment matrix $M = CD$ as a primary analytical tool. Through this method, they demonstrate that the non-trivial entries of the bidiagonal matrices are not merely arbitrary values but are explicitly linked to the coefficients of Type I and Type II multiple orthogonal polynomials. Specifically, these entries are expressed in terms of the coefficients of polynomials on the step-line and their corresponding Christoffel transformations. This connection allows for a profound translation of the matrix’s algebraic structure into the language of polynomial coefficients, providing a concrete way to represent matrix elements through polynomial properties.

Furthermore, the paper establishes a significant link between the production matrix of MOPs and the theory of branched continued fractions. By identifying that the non-diagonal elements of the Hessenberg matrix $H$ correspond precisely to the coefficients of the associated branched continued fraction, the authors derive a definitive criterion for factorization. They show that the existence of a bidiagonal factorization is equivalent to the condition that all coefficients of the branched continued fraction are non-zero and satisfy specific sign constraints. This leads to the important theoretical result that an oscillatory matrix $H$ (where all relevant coefficients are positive) inherently possesses a positive bidiagonal factorization, bridging matrix theory and continued fraction theory.

To validate the theoretical framework, the paper provides an explicit bidiagonal factorization for the Jacobi-Piñeiro polynomials. The authors also explore the limiting case where $r \to \infty$, demonstrating that the structure remains consistent as the system converges to the multiple Laguerre polynomials of the first kind. Ultimately, this work provides a powerful new set of mathematical tools for studying the spectral properties of multiple orthogonal polynomials and offers significant implications for the advanced modeling of stochastic processes, such as Markov chains, by providing a systematic way to decompose complex recurrence structures.