The non-symmetric Wilson polynomials are the Bannai-Ito polynomials

TL;DR

This paper demonstrates that non-symmetric Wilson polynomials are equivalent to Bannai-Ito polynomials, establishing algebraic isomorphisms and exploring their orthogonality and representations.

Contribution

It identifies the non-symmetric Wilson polynomials with Bannai-Ito polynomials and links their algebraic structures, including the degenerate double affine Hecke algebra and Bannai-Ito algebra.

Findings

Bannai-Ito polynomials satisfy an orthogonality relation with a positive measure.

An isomorphism between the degenerate double affine Hecke algebra and the Bannai-Ito algebra is established.

A non-compact form of the Bannai-Ito algebra with infinite-dimensional representations is introduced.

Abstract

The one-variable non-symmetric Wilson polynomials are shown to coincide with the Bannai-Ito polynomials. The isomorphism between the corresponding degenerate double affine Hecke algebra of type $(C_1^{\vee}, C_1)$ and the Bannai-Ito algebra is established. The Bannai-Ito polynomials are seen to satisfy an orthogonality relation with respect to a positive-definite and continuous measure on the real line. A non-compact form of the Bannai-Ito algebra is introduced and a four-parameter family of its infinite-dimensional and self-adjoint representations is exhibited.