Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
Abstract
We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.
Related
- Piñeiro mixed orthogonal polynomials and bidiagonal factorization
- Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions
- Positive Bidiagonal Factorizations for Banded Markov Processes
- The proportion of polynomials over F_2 with positive orthogonal multiplicity
- Orthogonal Quantum Latin Squares with Maximal Cardinality from Full-Rank Factorizations
BibTeX
@misc{manas2026positive,
title = {{Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials}},
author = {Manuel Mañas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08568},
eprint = {2610.08568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
AU - Manuel Mañas
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08568
UR - https://arxiv.org/abs/2610.08568
AB - We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.
ER -