Para families of orthogonal polynomials

Stéphane Z. Beaulac; Nicolas Crampé; Quentin Labriet; Lucia Morey; Rafael I. Nepomechie; Luc Vinet

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arXiv · 2609.38211

Para families of orthogonal polynomials

Stéphane Z. Beaulac·Nicolas Crampé·Quentin Labriet·Lucia Morey·Rafael I. Nepomechie·Luc Vinet

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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 $λ_n=λ_{N-n}$ 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.

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Stéphane Z. Beaulac, Nicolas Crampé, Quentin Labriet, Lucia Morey, Rafael I. Nepomechie, Luc Vinet. 2026-09-24. Para families of orthogonal polynomials. https://arxiv.org/abs/2609.38211

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