Para families of orthogonal polynomials
Abstract
The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, q-para-Racah and para-Bannai--Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras --- Hahn, Racah, q-Racah, Bannai--Ito and complementary Bannai--Ito --- are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing λn=λN−n of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.
Related
- Addition formula for two families of orthogonal polynomials in several variables
- The manufacture of examples II: the "para-classical" orthogonal polynomials
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
- An Exploration of Lacunary Orthogonal Polynomials Associated with the Geronimus Transformation
BibTeX
@misc{beaulac2026para,
title = {{Para families of orthogonal polynomials}},
author = {Stéphane Z. Beaulac and Nicolas Crampé and Quentin Labriet and Lucia Morey and Rafael I. Nepomechie and Luc Vinet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38211},
eprint = {2609.38211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38211},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Para families of orthogonal polynomials
AU - Stéphane Z. Beaulac
AU - Nicolas Crampé
AU - Quentin Labriet
AU - Lucia Morey
AU - Rafael I. Nepomechie
AU - Luc Vinet
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.38211
UR - https://arxiv.org/abs/2609.38211
AB - The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, q-para-Racah and para-Bannai--Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras --- Hahn, Racah, q-Racah, Bannai--Ito and complementary Bannai--Ito --- are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing λ_n=λ_N−n of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.
ER -