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

SIGMA 10 (2014), 038, 18 pages      arXiv:1309.7235      https://doi.org/10.3842/SIGMA.2014.038

A ''Continuous'' Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials

Vincent X. Genest a, Luc Vinet a and Alexei Zhedanov b
a) Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, QC, Canada, H3C 3J7
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received December 23, 2013, in final form March 24, 2014; Published online March 30, 2014

Abstract
A novel family of $-1$ orthogonal polynomials called the Chihara polynomials is characterized. The polynomials are obtained from a ''continuous'' limit of the complementary Bannai-Ito polynomials, which are the kernel partners of the Bannai-Ito polynomials. The three-term recurrence relation and the explicit expression in terms of Gauss hypergeometric functions are obtained through a limit process. A one-parameter family of second-order differential Dunkl operators having these polynomials as eigenfunctions is also exhibited. The quadratic algebra with involution encoding this bispectrality is obtained. The orthogonality measure is derived in two different ways: by using Chihara's method for kernel polynomials and, by obtaining the symmetry factor for the one-parameter family of Dunkl operators. It is shown that the polynomials are related to the big $-1$ Jacobi polynomials by a Christoffel transformation and that they can be obtained from the big $q$-Jacobi by a $q\rightarrow -1$ limit. The generalized Gegenbauer/Hermite polynomials are respectively seen to be special/limiting cases of the Chihara polynomials. A one-parameter extension of the generalized Hermite polynomials is proposed.

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

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