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

TL;DR

This paper explores properties of $q$-hypergeometric orthogonal polynomials at $q=-1$, introduces new polynomial classes, and provides matrix realizations of related algebraic structures.

Contribution

It characterizes a class of $q=-1$ orthogonal polynomials, constructs complementary classes via Darboux transformation, and introduces new examples and algebraic realizations.

Findings

Contains Bannai-Ito and related polynomials

Introduces new $-1$ polynomial examples

Provides matrix realizations of Bannai-Ito algebra

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.