Para-Bannai-Ito Polynomials

TL;DR

This paper introduces new bispectral polynomials called para-Bannai-Ito polynomials, orthogonal on a Bannai-Ito bi-lattice, characterized by recurrence, difference equations, and hypergeometric expressions, expanding the family of orthogonal polynomials.

Contribution

The paper constructs and characterizes para-Bannai-Ito polynomials from Bannai-Ito polynomials, including their explicit formulas and orthogonality, and links them to $q$-para-Racah and dual $-1$ Hahn polynomials.

Findings

Derived from Bannai-Ito polynomials via unconventional truncation.

Established orthogonality and recurrence relations.

Connected to $q$-para-Racah and dual $-1$ Hahn polynomials.

Abstract

New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform quadri-lattice) are obtained from an unconventional truncation of the untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete characterization of the resulting para-Bannai-Ito polynomials is provided, including a three term recurrence relation, a Dunkl-difference equation, an explicit expression in terms of hypergeometric series and an orthogonality relation. They are also derived as a $q\to -1$ limit of the $q$-para-Racah polynomials. A connection to the dual $-1$ Hahn polynomials is also established.