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

SIGMA 19 (2023), 090, 19 pages      arXiv:2209.10725      https://doi.org/10.3842/SIGMA.2023.090

Para-Bannai-Ito Polynomials

Jonathan Pelletier a, Luc Vinet ab and Alexei Zhedanov c
a) Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station,Montréal (Québec), H3C 3J7, Canada
b) IVADO, Montréal (Québec), H2S 3H1, Canada
c) School of Mathematics, Renmin University of China, Beijing 100872, P.R. China

Received June 10, 2023, in final form October 28, 2023; Published online November 10, 2023

Abstract
New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform quadri-lattice) are obtained from an unconventional truncation of the untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete characterization of the resulting para-Bannai-Ito polynomials is provided, including a three term recurrence relation, a Dunkl-difference equation, an explicit expression in terms of hypergeometric series and an orthogonality relation. They are also derived as a $q\to -1$ limit of the $q$-para-Racah polynomials. A connection to the dual $-1$ Hahn polynomials is also established.

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

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