Dual $-1$ Hahn polynomials: “Classical” polynomials beyond the Leonard duality
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- by Satoshi Tsujimoto, Luc Vinet and Alexei Zhedanov PDF
- Proc. Amer. Math. Soc. 141 (2013), 959-970 Request permission
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.References
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Additional Information
- Satoshi Tsujimoto
- Affiliation: Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-ku, Kyoto 606–8501, Japan
- MR Author ID: 339527
- Luc Vinet
- Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7, Canada
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Alexei Zhedanov
- Affiliation: Donetsk Institute for Physics and Technology, 83114 Donetsk, Ukraine
- MR Author ID: 234560
- Received by editor(s): July 31, 2011
- Published electronically: July 26, 2012
- Communicated by: Sergei K. Suslov
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 959-970
- MSC (2010): Primary 33C45
- DOI: https://doi.org/10.1090/S0002-9939-2012-11469-8
- MathSciNet review: 3003688