The Rahman polynomials and the Lie algebra
Authors:
Plamen Iliev and Paul Terwilliger
Journal:
Trans. Amer. Math. Soc. 364 (2012), 42254238
MSC (2010):
Primary 33C52; Secondary 17B10, 33C45, 33D45
Published electronically:
March 20, 2012
MathSciNet review:
2912452
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Abstract: We interpret the Rahman polynomials in terms of the Lie algebra . Using the parameters of the polynomials we define two Cartan subalgebras for , denoted and . We display an antiautomorphism of that fixes each element of and each element of . We consider a certain finitedimensional irreducible module consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form on such that for all and . We display two bases for ; one diagonalizes and the other diagonalizes . Both bases are orthogonal with respect to . We show that when 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 seventerm 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 and are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of and on 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 303320160
Email:
iliev@math.gatech.edu
Paul Terwilliger
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 537061388
Email:
terwilli@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S00029947201205495X
PII:
S 00029947(2012)05495X
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 DMS0901092.
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
