2012/11/30 by Vincent X. Genest, Luc Vinet, Alexei Zhedanov
Mathematics · Physics and Astronomy · #Algebra over a field #Classical orthogonal polynomials #Difference polynomials #Discrete orthogonal polynomials #Eigenfunction #Eigenvalues and eigenvectors #Gegenbauer polynomials #Geometry #Hahn polynomials #Jacobi polynomials #Kravchuk polynomials #Mathematical functions and polynomials #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Physics #Pure mathematics #Quadratic equation #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Wilson polynomials #math-ph #math.CA #math.MP
paper · pdf · doi:10.3842/sigma.2013.018
published as SIGMA 9 (2013), 018, 20 pages
arxiv created 2013/03/02 · openalex publication_date 2013/03/02 · arxiv updated 2013/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 -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.