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.
 1.
R.
W. Carter, Lie algebras of finite and affine type, Cambridge
Studies in Advanced Mathematics, vol. 96, Cambridge University Press,
Cambridge, 2005. MR 2188930
(2006i:17001)
 2.
J.
J. Duistermaat and F.
A. Grünbaum, Differential equations in the spectral
parameter, Comm. Math. Phys. 103 (1986), no. 2,
177–240. MR
826863 (88j:58106)
 3.
Jeffrey
S. Geronimo and Plamen
Iliev, Bispectrality of multivariable RacahWilson
polynomials, Constr. Approx. 31 (2010), no. 3,
417–457. MR 2608419
(2011m:39028), 10.1007/s0036500990453
 4.
F.
Alberto Grünbaum, The Rahman polynomials are bispectral,
SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007),
Paper 065, 11. MR 2322792
(2009m:33019), 10.3842/SIGMA.2007.065
 5.
Michael
R. Hoare and Mizan
Rahman, A probabilistic origin for a new class of bivariate
polynomials, SIGMA Symmetry Integrability Geom. Methods Appl.
4 (2008), Paper 089, 18. MR 2470507
(2010a:33028), 10.3842/SIGMA.2008.089
 6.
Plamen
Iliev, Bispectral commuting difference
operators for multivariable AskeyWilson polynomials, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1577–1598. MR 2737278
(2011j:42050), 10.1090/S000299472010051839
 7.
Tatsuro
Ito, Kazumasa
Nomura, and Paul
Terwilliger, A classification of sharp tridiagonal pairs,
Linear Algebra Appl. 435 (2011), no. 8,
1857–1884. MR 2810633
(2012e:15027), 10.1016/j.laa.2011.03.032
 8.
Tatsuro
Ito, Kenichiro
Tanabe, and Paul
Terwilliger, Some algebra related to 𝑃 and
𝑄polynomial association schemes, Codes and association
schemes (Piscataway, NJ, 1999) DIMACS Ser. Discrete Math. Theoret.
Comput. Sci., vol. 56, Amer. Math. Soc., Providence, RI, 2001,
pp. 167–192. MR 1816397
(2002h:05162)
 9.
Jens
Carsten Jantzen, Lectures on quantum groups, Graduate Studies
in Mathematics, vol. 6, American Mathematical Society, Providence, RI,
1996. MR
1359532 (96m:17029)
 10.
R. Koekoek and R. F. Swarttouw.
The Askey scheme of hypergeometric orthogonal polyomials and its analog, report 9817, Delft University of Technology, The Netherlands, 1998. Available at http://fa.its.tudelft.nl/koekoek/askey.html
 11.
Hiroshi
Mizukawa and Hajime
Tanaka, (𝑛+1,𝑚+1)hypergeometric
functions associated to character algebras, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2613–2618 (electronic). MR 2054786
(2005h:33022), 10.1090/S000299390407399X
 12.
Paul
Terwilliger, Two linear transformations each tridiagonal with
respect to an eigenbasis of the other, Linear Algebra Appl.
330 (2001), no. 13, 149–203. MR 1826654
(2002h:15021), 10.1016/S00243795(01)002427
 13.
Paul
Terwilliger, An algebraic approach to the Askey scheme of
orthogonal polynomials, Orthogonal polynomials and special functions,
Lecture Notes in Math., vol. 1883, Springer, Berlin, 2006,
pp. 255–330. MR 2243532
(2007g:33011), 10.1007/9783540367161_6
 1.
 R. Carter.
Lie algebras of finite and affine type. Cambridge Studies in Advanced Mathematics 96. Cambridge U. Press, Cambridge, 2005. MR 2188930 (2006i:17001)
 2.
 J. J. Duistermaat and F. A. Grünbaum.
Differential equations in the spectral parameter. Comm. Math. Phys. 103 (1986), 177240. MR 826863 (88j:58106)
 3.
 J. Geronimo and P. Iliev.
Bispectrality of multivariable RacahWilson polynomials. Constr. Approx. 31 (2010), 417457. MR 2608419
 4.
 F. A. Grünbaum.
The Rahman polynomials are bispectral. SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007) Paper 065, 11 pp. (electronic). MR 2322792 (2009m:33019)
 5.
 M.R. Hoare and M. Rahman.
A probablistic origin for a new class of bivariate polynomials. SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008) Paper 089, 18 pp. (electronic). MR 2470507 (2010a:33028)
 6.
 P. Iliev.
Bispectral commuting difference operators for multivariable AskeyWilson polynomials. Trans. Amer. Math. Soc. 363 (2011), no. 3, 15771598. MR 2737278
 7.
 T. Ito, K. Nomura, P. Terwilliger.
A classification of sharp tridiagonal pairs. Linear Algebra Appl.. 435 (2011), 18571884. MR 2810633
 8.
 T. Ito, K. Tanabe, and P. Terwilliger.
Some algebra related to  and polynomial association schemes, in: Codes and Association Schemes (Piscataway NJ, 1999), Amer. Math. Soc., Providence RI, 2001, pp. 167192. MR 1816397 (2002h:05162)
 9.
 J. Jantzen.
Lectures on quantum groups. Graduate Studies in Mathematics, 6. Amer. Math. Soc. Providence, RI, 1996. MR 1359532 (96m:17029)
 10.
 R. Koekoek and R. F. Swarttouw.
The Askey scheme of hypergeometric orthogonal polyomials and its analog, report 9817, Delft University of Technology, The Netherlands, 1998. Available at http://fa.its.tudelft.nl/koekoek/askey.html
 11.
 H. Mizukawa and H. Tanaka.
hypergeometric functions associated to character algebras. Proc. Amer. Math. Soc. 132 (2004), 26132618. MR 2054786 (2005h:33022)
 12.
 P. Terwilliger.
Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebra Appl. 330 (2001), 149203. MR 1826654 (2002h:15021)
 13.
 P. Terwilliger.
An algebraic approach to the Askey scheme of orthogonal polynomials. Orthogonal polynomials and special functions, 255330, Lecture Notes in Math., 1883, Springer, Berlin, 2006. MR 2243532 (2007g:33011)
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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
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.
