An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality
Abstract
We embed the Jacobi-like and Laguerre-like systems for ordinary type-I/type-II multiple orthogonality introduced by Wolfs into a single Askey-type confluence scheme. Applying the Bernstein transform simultaneously to the q Jacobi-like weights produces a positive finite-lattice Hahn-like ancestor, represented by multiple beta integrals, reducing to the classical Hahn weight for q=1, and converging back to the continuous system. Parameter and scaling limits then yield Kravchuk-like, two Meixner-like, two Charlier-like, Jacobi-like, two Laguerre-like, and Hermite-like families. We realize every arrow through explicit hypergeometric orthogonality data. For each near-diagonal multi-index 1≤ |m|≤ N, the Hahn-like type-II polynomial is a terminating q+2Fq+1(1) series satisfying an exact inverse Bernstein identity. We construct the normalized type-I form and, under explicit separation and nonvanishing assumptions, recover its individual polynomial components by finite sums of terminating hypergeometric functions and prove uniqueness. Finite-pole terms regroup into terminating Kampé de Fériet blocks. Their limits are obtained by coefficient extraction and reconstruction at infinity on the unreflected Kravchuk-like and Meixner-II-like branches, and by finite Lauricella--Horn sector sums on the reflected branches. This yields sectorwise component confluence along the Kravchuk--Charlier, Meixner--Charlier, Meixner--Laguerre, and Charlier--Hermite arrows. For q=1 all formulas reduce to the classical families; for q>1 the two Meixner-like systems, and likewise the two Charlier-like systems, are inequivalent under permutation of normalized rows.
Related
- An Askey Scheme for Jacobi Systems of Mixed Type: Laguerre Limits of the First and Second Kinds and Hermite Limits
- Confluence relations for q-analogues of multiple zeta values
- Bernstein-type bases on q-quadratic lattices and Askey--Wilson\slash q-Racah connection coefficients
- James-type constants for Birkhoff--James orthogonality and approximate isosceles
- Hahn-Type Orthogonality for the Resolvent Delta Operator ℒ=D(I−D)−1
BibTeX
@misc{manas2026askeytype,
title = {{An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality}},
author = {Manuel Mañas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08048},
eprint = {2609.08048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08048},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Askey-Type Confluence Scheme for Hahn-Like Multiple Orthogonality
AU - Manuel Mañas
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.08048
UR - https://arxiv.org/abs/2609.08048
AB - We embed the Jacobi-like and Laguerre-like systems for ordinary type-I/type-II multiple orthogonality introduced by Wolfs into a single Askey-type confluence scheme. Applying the Bernstein transform simultaneously to the q Jacobi-like weights produces a positive finite-lattice Hahn-like ancestor, represented by multiple beta integrals, reducing to the classical Hahn weight for q=1, and converging back to the continuous system. Parameter and scaling limits then yield Kravchuk-like, two Meixner-like, two Charlier-like, Jacobi-like, two Laguerre-like, and Hermite-like families. We realize every arrow through explicit hypergeometric orthogonality data. For each near-diagonal multi-index 1≤ |m|≤ N, the Hahn-like type-II polynomial is a terminating _q+2F_q+1(1) series satisfying an exact inverse Bernstein identity. We construct the normalized type-I form and, under explicit separation and nonvanishing assumptions, recover its individual polynomial components by finite sums of terminating hypergeometric functions and prove uniqueness. Finite-pole terms regroup into terminating Kampé de Fériet blocks. Their limits are obtained by coefficient extraction and reconstruction at infinity on the unreflected Kravchuk-like and Meixner-II-like branches, and by finite Lauricella--Horn sector sums on the reflected branches. This yields sectorwise component confluence along the Kravchuk--Charlier, Meixner--Charlier, Meixner--Laguerre, and Charlier--Hermite arrows. For q=1 all formulas reduce to the classical families; for q>1 the two Meixner-like systems, and likewise the two Charlier-like systems, are inequivalent under permutation of normalized rows.
ER -