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

Luis Verde-Star

TL;DR

The paper develops a unified framework for $q$-hypergeometric orthogonal polynomials at $q=-1$ by introducing a parametric class $\mathcal{A}$ with a uniform recurrence structure and a complementary class $\mathcal{C}$ generated by a Darboux shift. It shows that the Bannai--Ito polynomials and their companions arise naturally within these families and presents new $-1$ polynomial examples, along with matrix realizations of the Bannai--Ito algebra. Through systematic parameter reductions (notably for $b_2\neq 0$ and $b_2=0$) and the construction of a tractable subclass $\mathcal{C}_0$, the work embeds known $-1$ polynomials and reveals simple recurrence patterns. The study also establishes several infinite-matrix realizations of the Bannai--Ito algebra and discusses potential extensions toward a $-1$ Askey scheme and related open problems, offering a solid algebraic and analytic foundation for further exploration of $-1$ polynomials in hypergeometric families.

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.

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

TL;DR

The paper develops a unified framework for -hypergeometric orthogonal polynomials at by introducing a parametric class with a uniform recurrence structure and a complementary class generated by a Darboux shift. It shows that the Bannai--Ito polynomials and their companions arise naturally within these families and presents new polynomial examples, along with matrix realizations of the Bannai--Ito algebra. Through systematic parameter reductions (notably for and ) and the construction of a tractable subclass , the work embeds known polynomials and reveals simple recurrence patterns. The study also establishes several infinite-matrix realizations of the Bannai--Ito algebra and discusses potential extensions toward a Askey scheme and related open problems, offering a solid algebraic and analytic foundation for further exploration of polynomials in hypergeometric families.

Abstract

We obtain some properties of a class of -hypergeometric orthogonal polynomials with , described by a uniform parametrization of the recurrence coefficients. We construct a class of complementary 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 polynomials. We introduce some new examples of polynomials and also obtain matrix realizations of the Bannai-Ito algebra.

Paper Structure

This paper contains 10 sections, 97 equations.

Table of Contents

  1. Introduction
  2. Construction of bispectral hypergeometric orthogonal polynomials
  3. The polynomials with $\boldsymbol{q=-1}$
  4. A change of parameters for the case with $\boldsymbol{b_2 \ne 0}$
  5. A parametrization for the $\boldsymbol{-1}$ polynomials with $\boldsymbol{b_2=0}$
  6. The class of complementary $\boldsymbol{-1}$ polynomials
  7. The subclass $\boldsymbol{\mathcal{C}_0}$
  8. Another parametrization of the subclass $\boldsymbol{\mathcal{C}_0}$
  9. The Bannai--Ito algebra
  10. Final remarks

Theorems & Definitions (3)

  • Example 5.1
  • Example 5.2
  • Example 5.3