Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 018, 20 pages      arXiv:1211.2461      https://doi.org/10.3842/SIGMA.2013.018

Bispectrality of the Complementary Bannai-Ito Polynomials

Vincent X. Genest a, Luc Vinet a and Alexei Zhedanov b
a) Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal, Québec, Canada, H3C 3J7
b) Donetsk Institute for Physics and Technology, Ukraine

Received November 13, 2012, in final form February 27, 2013; Published online March 02, 2013

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→−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.

Key words: Bannai-Ito polynomials; quadratic algebras; Dunkl operators.

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