Bispectrality of the Complementary Bannai–Ito Polynomials

A one-parameter family of operators that have the complementary Bannai–Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai–Ito polynomials and also correspond to a $q\rightarrow-1$ limit of the Askey–Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey–Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual $-1$ Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.