[Paper Review] Dunkl shift operators and Bannai-Ito polynomials

Satoshi Tsujimoto, Luc Vinet+1 more|arXiv (Cornell University)|Jun 17, 2011

Mathematical functions and polynomialsMathematics20 references19 citations

TL;DR

This paper introduces a general class of first-order Dunkl shift operators involving shift and reflection operators that preserve polynomial spaces and are potentially self-adjoint. It demonstrates that the eigenpolynomials of such operators are the Bannai-Ito polynomials, providing a new eigenvalue characterization, algebraic structure via a q = -1 AW(3) algebra, and a new explicit expression using complementary Bannai-Ito polynomials related to Wilson polynomials.

ABSTRACT

We consider the most general Dunkl shift operator $L$ with the following properties: (i) $L$ is of first order in the shift operator and involves reflections; (ii) $L$ preserves the space of polynomials of a given degree; (iii) $L$ is potentially self-adjoint. We show that under these conditions, the operator $L$ has eigenfunctions which coincide with the Bannai-Ito polynomials. We construct a polynomial basis which is lower-triangular and two-diagonal with respect to the action of the operator $L$. This allows to express the BI polynomials explicitly. We also present an anti-commutator AW(3) algebra corresponding to this operator. From the representations of this algebra, we derive the structure and recurrence relations of the BI polynomials. We introduce new orthogonal polynomials - referred to as the complementary BI polynomials - as an alternative $q o -1$ limit of the Askey-Wilson polynomials. These complementary BI polynomials lead to a new explicit expression for the BI polynomials in terms of the ordinary Wilson polynomials.

Motivation & Objective

• To characterize the most general first-order Dunkl shift operator that preserves polynomial spaces and is potentially self-advantageous.
• To show that the eigenpolynomials of such operators coincide with the Bannai-Ito polynomials, providing a new eigenvalue equation for them.
• To construct a lower-triangular, two-diagonal basis that enables explicit expressions for the Bannai-Ito polynomials.
• To introduce a q = -1 version of the AW(3) algebra (called the Bannai-Ito algebra) and use it to derive recurrence and structure relations for the polynomials.
• To define complementary Bannai-Ito polynomials as a distinct q → -1 limit of Askey-Wilson polynomials and use them to express Bannai-Ito polynomials in terms of Wilson polynomials.

Proposed method

• The authors define a Dunkl shift operator L involving the shift operator T⁺ and reflection operator R, with rational coefficients depending on parameters.
• They impose conditions ensuring L maps polynomials to polynomials of the same degree, enabling the existence of polynomial eigenfunctions.
• A polynomial basis {ϕₙ(x)} is constructed such that Lϕₙ = λₙϕₙ + νₙϕₙ₋₁, making L lower-triangular with two diagonals.
• The eigenpolynomials Pₙ(x) are derived as linear combinations of hypergeometric functions ₄F₃(1) via this basis.
• The algebraic structure is revealed by showing that L and multiplication by x generate a q = -1 analogue of the AW(3) algebra, now termed the Bannai-Ito algebra.
• Complementary Bannai-Ito polynomials are introduced as a distinct q → -1 limit of Askey-Wilson polynomials, and their Christoffel transform property is used to express BI polynomials as linear combinations of Wilson polynomials.

Experimental results

Research questions

• RQ1 What is the most general first-order Dunkl shift operator involving shift and reflection that preserves polynomial spaces and is potentially self-adjoint?
• RQ2 Do the eigenpolynomials of such an operator coincide with the Bannai-Ito polynomials, and can they be explicitly constructed?
• RQ3 Can the Bannai-Ito polynomials be derived algebraically from a q = -1 version of the AW(3) algebra?
• RQ4 How do the complementary Bannai-Ito polynomials relate to the standard Bannai-Ito polynomials and to Wilson polynomials?
• RQ5 What are the limiting cases of the Bannai-Ito polynomials, and how do they relate to known classical orthogonal polynomials?

Key findings

• The eigenpolynomials of the general Dunkl shift operator L are identified as the Bannai-Ito polynomials, providing a new eigenvalue equation involving shift and reflection operators.
• A lower-triangular, two-diagonal basis {ϕₙ(x)} is constructed, enabling explicit expressions for the Bannai-Ito polynomials as linear combinations of ₄F₃(1) hypergeometric functions.
• The operator L and multiplication by x generate a q = -1 version of the AW(3) algebra, now called the Bannai-Ito algebra, which allows derivation of the standard three-term recurrence relation for the Bannai-Ito polynomials.
• The complementary Bannai-Ito polynomials are introduced as a new q → -1 limit of Askey-Wilson polynomials, distinct from the one used by Bannai and Ito, and are shown to be Christoffel transforms of the Bannai-Ito polynomials.
• The Bannai-Ito polynomials are expressed as a linear combination of two Wilson polynomials via the complementary polynomials, providing a new explicit formula.
• The symmetric Bannai-Ito polynomials are shown to include the symmetric Meixner-Pollaczek polynomials as a special case, with weight function |Γ(a + 2ix)|².