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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Rahman polynomials and the Lie algebra $ \mathfrak{sl}_3(\mathbb{C})$


Authors: Plamen Iliev and Paul Terwilliger
Journal: Trans. Amer. Math. Soc. 364 (2012), 4225-4238
MSC (2010): Primary 33C52; Secondary 17B10, 33C45, 33D45
Published electronically: March 20, 2012
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Abstract: We interpret the Rahman polynomials in terms of the Lie algebra $ \mathfrak{sl}_3(\mathbb{C})$. Using the parameters of the polynomials we define two Cartan subalgebras for $ \mathfrak{sl}_3(\mathbb{C})$, denoted $ H$ and $ \tilde H$. We display an antiautomorphism $ \dagger $ of $ \mathfrak{sl}_3(\mathbb{C})$ that fixes each element of $ H$ and each element of $ \tilde H$. We consider a certain finite-dimensional irreducible $ \mathfrak{sl}_3(\mathbb{C})$-module $ V$ consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form $ \langle \,,\,\rangle $ on $ V$ such that $ \langle \beta \xi ,\zeta \rangle = \langle \xi ,\beta ^\dagger \zeta \rangle $ for all $ \beta \in \mathfrak{sl}_3(\mathbb{C})$ and $ \xi ,\zeta \in V$. We display two bases for $ V$; one diagonalizes $ H$ and the other diagonalizes $ \tilde H$. Both bases are orthogonal with respect to $ \langle \,,\,\rangle $. We show that when $ \langle \,,\,\rangle $ is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we recover the previously known orthogonality relation for the Rahman polynomials. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by the dual polynomials. These recurrence relations show that the Rahman polynomials are bispectral. In our theory the roles of $ H$ and $ \tilde H$ are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of $ H$ and $ \tilde H$ on $ V$ as a rank 2 generalization of a Leonard pair.


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Additional Information

Plamen Iliev
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: iliev@math.gatech.edu

Paul Terwilliger
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: terwilli@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05495-X
PII: S 0002-9947(2012)05495-X
Keywords: Orthogonal polynomial, Askey scheme, Leonard pair, tridiagonal pair.
Received by editor(s): June 24, 2010
Received by editor(s) in revised form: October 31, 2010
Published electronically: March 20, 2012
Additional Notes: The first author was supported in part by NSF grant DMS-0901092.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.