Bispectrality of the Complementary Bannai-Ito Polynomials

Abstract

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 $q\to -1$ 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.

References in corpus (8)

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• Askey-Wilson relations and Leonard pairs
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Cited by in corpus (20)

• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• The Dunkl oscillator in three dimensions
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The Hahn superalgebra and supersymmetric Dunkl oscillator models
• A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials
• The ${\mathbb{Z}}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra
• The Heun-Racah and Heun-Bannai-Ito algebras
• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Generating functions for the osp(1|2) Clebsch-Gordan coefficients
• Bivariate Bannai-Ito polynomials
• Superintegrability and the dual $-1$ Hahn algebra in superconformal quantum mechanics
• A superintegrable model with reflections on $S^3$ and the rank two Bannai-Ito algebra
• Orthogonal polynomials and the deformed Jordan plane
• Para-Bannai-Ito Polynomials
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra
• Symmetric abstract hypergeometric polynomials