Bessel Multiple Orthogonal Polynomials of Mixed Type
Abstract
This article constructs a Bessel family of mixed-type multiple orthogonal polynomials for a q× p matrix weight on the unit circle. Unlike a rank-one product weight, this matrix has generic rank min{q,p} outside a finite subset of the circle. Its reciprocal-Gamma moments recover the multiple Bessel system when q=1 and the Bessel-like system of Wolfs when p=1. The same matrix is obtained as a scaled Markov-Stieltjes limit of a rank-one Jacobi system, although the interval measures themselves have no finite limit. For balanced near-diagonal indices, explicit formulas are obtained for the mixed A and B polynomial vectors. Their orthogonality and weak normality are proved, and componentwise strong normality is characterized. The components have terminating generalized hypergeometric representations; the B components also admit finite Kampé de Fériet representations and a matrix Rodrigues-type formula. In the one-row reduction, the bivariate representation becomes a generalized hypergeometric polynomial governed by a reflected type-II multiple Hahn polynomial. Finite Gamma-Pochhammer formulas give the near-diagonal and step-line recurrence coefficients. The corresponding banded recurrence matrix has a bidiagonal Christoffel factorization: the lower factors are evaluated from transformed polynomial vectors, while the upper factors are expressed through finite tau determinants. When q=1, every Christoffel step remains within the multiple Bessel family, and Gamma-Vandermonde determinants yield the complete factorization.
Related
- Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
- Hypergeometric Mixed-Type Multiple Orthogonal Polynomials
- Piñeiro mixed orthogonal polynomials and bidiagonal factorization
- A method for generating multivariate multiple orthogonal polynomials
- Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials
BibTeX
@misc{manas2026bessel,
title = {{Bessel Multiple Orthogonal Polynomials of Mixed Type}},
author = {Manuel Mañas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.00781},
eprint = {2608.00781},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00781},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bessel Multiple Orthogonal Polynomials of Mixed Type
AU - Manuel Mañas
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2608.00781
UR - https://arxiv.org/abs/2608.00781
AB - This article constructs a Bessel family of mixed-type multiple orthogonal polynomials for a q× p matrix weight on the unit circle. Unlike a rank-one product weight, this matrix has generic rank min{q,p} outside a finite subset of the circle. Its reciprocal-Gamma moments recover the multiple Bessel system when q=1 and the Bessel-like system of Wolfs when p=1. The same matrix is obtained as a scaled Markov-Stieltjes limit of a rank-one Jacobi system, although the interval measures themselves have no finite limit. For balanced near-diagonal indices, explicit formulas are obtained for the mixed A and B polynomial vectors. Their orthogonality and weak normality are proved, and componentwise strong normality is characterized. The components have terminating generalized hypergeometric representations; the B components also admit finite Kampé de Fériet representations and a matrix Rodrigues-type formula. In the one-row reduction, the bivariate representation becomes a generalized hypergeometric polynomial governed by a reflected type-II multiple Hahn polynomial. Finite Gamma-Pochhammer formulas give the near-diagonal and step-line recurrence coefficients. The corresponding banded recurrence matrix has a bidiagonal Christoffel factorization: the lower factors are evaluated from transformed polynomial vectors, while the upper factors are expressed through finite tau determinants. When q=1, every Christoffel step remains within the multiple Bessel family, and Gamma-Vandermonde determinants yield the complete factorization.
ER -