Hypergeometric Mixed-Type Multiple Orthogonal Polynomials
Abstract
Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one q× p matrices of weights, with arbitrary p and q: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at (x,1). For the mixed beta--Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer G-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with p subdiagonals and q superdiagonals. All recurrence coefficients are given by finite Gamma--Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss--Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.
Related
- Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
- Bessel Multiple Orthogonal Polynomials of Mixed Type
- Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients
- Piñeiro mixed orthogonal polynomials and bidiagonal factorization
- A method for generating multivariate multiple orthogonal polynomials
BibTeX
@misc{manas2026hypergeometric,
title = {{Hypergeometric Mixed-Type Multiple Orthogonal Polynomials}},
author = {Manuel Mañas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26110},
eprint = {2609.26110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26110},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hypergeometric Mixed-Type Multiple Orthogonal Polynomials
AU - Manuel Mañas
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.26110
UR - https://arxiv.org/abs/2609.26110
AB - Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one q× p matrices of weights, with arbitrary p and q: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at (x,1). For the mixed beta--Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer G-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with p subdiagonals and q superdiagonals. All recurrence coefficients are given by finite Gamma--Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss--Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.
ER -