An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits
Abstract
An Askey-type confluence scheme is developed for Jacobi multiple orthogonal systems of mixed type, with q row weights and p column weights. Rescaling the Jacobi variable near $0$ and near $1$ produces Laguerre systems of the first and second kinds. A Hermite system with one Gaussian--gamma row and q−1 exponential convolutions is obtained from Laguerre of the first kind, directly from the second kind, and directly from Jacobi. Explicit limiting weights and orthogonal forms are given, together with convergence of their mixed moments. For each fixed index, the recurrence coefficients and the entries of the bidiagonal factors also converge in the limit to Laguerre of the first kind, under nonvanishing conditions. The Laguerre family of the second kind is described by additive gamma convolutions. A closed product formula for its moment determinants holds for arbitrary positive real shape parameters and near-diagonal row multi-indices, yielding explicit normality conditions. Its forms admit contour and Rodrigues representations with terminating hypergeometric components: a multiple Kampé de Fériet series for A, and at most q+1 Srivastava--Daoust series for each B-component when q≥2. For the Hermite family, normality holds at every Laguerre-admissible index under the stated separation conditions, and these indices form an infinite set. Its components have finite Hermite expansions and terminating Srivastava--Daoust representations. The three routes give the same limiting moments and normalized components, the Jacobi one being a single limit and the other two iterated. An appendix treats several derivative rows, where the connection determinant vanishes beyond a finite index range, with exact rank loss and diverging components. The one-row reductions recover the classical multiple second-kind Laguerre and Hermite systems.
Related
- Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem
- The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition
- From Hermite Chaos to an Attractive Log-Gas: Explicit High-Rank Limits of Iterated-Logarithm and Tail Constants
- Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems
- Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data
BibTeX
@misc{manas2026askey,
title = {{An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits}},
author = {Manuel Mañas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22027},
eprint = {2609.22027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22027},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits
AU - Manuel Mañas
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22027
UR - https://arxiv.org/abs/2609.22027
AB - An Askey-type confluence scheme is developed for Jacobi multiple orthogonal systems of mixed type, with q row weights and p column weights. Rescaling the Jacobi variable near $0$ and near $1$ produces Laguerre systems of the first and second kinds. A Hermite system with one Gaussian--gamma row and q−1 exponential convolutions is obtained from Laguerre of the first kind, directly from the second kind, and directly from Jacobi. Explicit limiting weights and orthogonal forms are given, together with convergence of their mixed moments. For each fixed index, the recurrence coefficients and the entries of the bidiagonal factors also converge in the limit to Laguerre of the first kind, under nonvanishing conditions. The Laguerre family of the second kind is described by additive gamma convolutions. A closed product formula for its moment determinants holds for arbitrary positive real shape parameters and near-diagonal row multi-indices, yielding explicit normality conditions. Its forms admit contour and Rodrigues representations with terminating hypergeometric components: a multiple Kampé de Fériet series for A, and at most q+1 Srivastava--Daoust series for each B-component when q≥2. For the Hermite family, normality holds at every Laguerre-admissible index under the stated separation conditions, and these indices form an infinite set. Its components have finite Hermite expansions and terminating Srivastava--Daoust representations. The three routes give the same limiting moments and normalized components, the Jacobi one being a single limit and the other two iterated. An appendix treats several derivative rows, where the connection determinant vanishes beyond a finite index range, with exact rank loss and diverging components. The one-row reductions recover the classical multiple second-kind Laguerre and Hermite systems.
ER -