
SIGMA 10 (2014), 038, 18 pages arXiv:1309.7235
https://doi.org/10.3842/SIGMA.2014.038
A ''Continuous'' Limit of the Complementary BannaiIto 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. CentreVille, 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 BannaiIto polynomials, which are the
kernel partners of the BannaiIto polynomials. The threeterm recurrence relation and the explicit expression in terms of Gauss hypergeometric functions are obtained
through a limit process. A oneparameter family of secondorder 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 oneparameter 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 oneparameter extension of the generalized Hermite polynomials is proposed.
Key words:
BannaiIto polynomials; Dunkl operators; orthogonal polynomials; quadratic algebras.
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