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

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.