Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 083, 19 pages      arXiv:2410.14068      https://doi.org/10.3842/SIGMA.2025.083

$q$-Hypergeometric Orthogonal Polynomials with $q=-1$

Luis Verde-Star
Department of Mathematics, Universidad Autónoma Metropolitana, Iztapalapa, Mexico City, Mexico

Received October 31, 2024, in final form September 18, 2025; Published online October 02, 2025

Abstract
We obtain some properties of a class $\mathcal{A}$ of $q$-hypergeometric orthogonal polynomials with $q=-1$, described by a uniform parametrization of the recurrence coefficients. We construct a class $\mathcal{C}$ of complementary $-1$ polynomials by means of the Darboux transformation with a shift. We show that our classes contain the Bannai-Ito polynomials and their complementary polynomials and other known $-1$ polynomials. We introduce some new examples of $-1$ polynomials and also obtain matrix realizations of the Bannai-Ito algebra.

Key words: hypergeometric orthogonal polynomials; recurrence coefficients; $-1$ orthogonal polynomials; Bannai-Ito polynomials; Bannai-Ito algebra.

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