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Dual $ -1$ Hahn polynomials: ``Classical'' polynomials beyond the Leonard duality

Authors: Satoshi Tsujimoto, Luc Vinet and Alexei Zhedanov
Journal: Proc. Amer. Math. Soc. 141 (2013), 959-970
MSC (2010): Primary 33C45
Published electronically: July 26, 2012
MathSciNet review: 3003688
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Abstract: We introduce the $ -$1 dual Hahn polynomials through an appropriate $ q \to -1$ limit of the dual $ q$-Hahn polynomials. These polynomials are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical orthogonal polynomials of the Askey scheme, the $ -1$ dual Hahn polynomials do not exhibit the Leonard duality property. Instead, these polynomials satisfy a 4th-order difference eigenvalue equation and thus possess a bispectrality property. The corresponding generalized Leonard pair consists of two matrices $ A,B$ each of size $ N+1 \times N+1$. In the eigenbasis where the matrix $ A$ is diagonal, the matrix $ B$ is 3-diagonal; but in the eigenbasis where the matrix $ B$ is diagonal, the matrix $ A$ is 5-diagonal.

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Satoshi Tsujimoto
Affiliation: Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-ku, Kyoto 606–8501, Japan

Luc Vinet
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7, Canada

Alexei Zhedanov
Affiliation: Donetsk Institute for Physics and Technology, 83114 Donetsk, Ukraine

Keywords: Classical orthogonal polynomials, dual $q$-Hahn polynomials, Leonard duality.
Received by editor(s): July 31, 2011
Published electronically: July 26, 2012
Communicated by: Sergei K. Suslov
Article copyright: © Copyright 2012 American Mathematical Society

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