Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

On $BC$ type basic hypergeometric orthogonal polynomials

Author(s): Jasper V. Stokman
Journal: Trans. Amer. Math. Soc. 352 (2000), 1527-1579.
MSC (2000): Primary 33D52; Secondary 33D45, 33D80
Posted: November 17, 1999
MathSciNet review: 1694379
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable $q$-Racah polynomials and multivariable big and little $q$-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known $q$-analogues of the Selberg integral are obtained.

References:

[1]
G.E. Andrews, R. Askey, Enumeration of partitions: the role of Eulerian series and $q$-orthogonal polynomials, in Higher Combinatorics, M. Aigner, ed., Reidel, Boston, MA (1977), pp. 1-99. MR 80b:10021
[2]
G.E. Andrews, R. Askey, Classical orthogonal polynomials, in Polynômes Orthogonaux et Applications, C. Brezinski, A. Draux, A.P. Magnus, P. Maroni and A. Ronveaux, eds., Lecture Notes in Math. 1171, Springer, New York (1985), pp. 1-99. MR 88c:33015b
[3]
K. Aomoto, On Elliptic Product Formulas for Jackson Integrals associated with Reduced Root Systems, J. of Alg. Comb. 8 (1998), pp. 1-99. CMP 99:02
[4]
R. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), pp. 1-99. MR 82e:33002
[5]
R. Askey, J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or $6$-j symbols, SIAM J. Math. Anal. 10 (1979), pp. 1-99. MR 80k:33012
[6]
R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319. MR 87a:05023
[7]
T.H. Baker, P.J. Forrester, Multivariable Al-Salam & Carlitz polynomials associated with the type A $q$-Dunkl kernel, preprint (1997).
[8]
I. Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. Math. 141 (1995), pp. 1-99. MR 96m:33010
[9]
I. Cherednik, Macdonald's evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), pp. 1-99. MR 98i:33027a
[10]
I. Cherednik, Nonsymmetric Macdonald polynomials, IMRN 10 (1995), pp. 1-99. MR 97f:33032
[11]
I. Cherednik, Intertwining operators of double affine Hecke algebras, Selecta Math. (N.S.) 3 (1997), pp. 1-99. MR 99g:33036
[12]
J.F. van Diejen, Self-dual Koornwinder-Macdonald polynomials, Invent. Math. 126 (1996), pp. 1-99. MR 98k:33025
[13]
J.F. van Diejen, On certain multiple Bailey, Rogers and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), pp. 1-99. MR 98j:33011
[14]
J.F. van Diejen, J.V. Stokman, Multivariable q-Racah polynomials, Duke Math. J. 91 (1998), pp. 89-136. MR 99b:33039
[15]
R.J. Evans, Multidimensional beta and gamma integrals, Contemp. Math. 166 (1994), pp. 1-99. MR 96e:33036
[16]
R.A. Gustafson, Some $q$-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras, Trans. Amer. Math. Soc. 341 (1994), pp. 1-99. MR 94c:33032
[17]
R.A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), pp. 1-99. MR 90j:33001
[18]
G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Math. and its Appl., vol. 35, Cambridge University Press, Cambridge (1990). MR 91d:33034
[19]
L. Habsieger, Une q-intégrale de Selberg et Askey, SIAM J. Math. Anal. 19 (1988), pp. 1-99. MR 89m:33002
[20]
M. Ito, On a theta product formula for Jackson integrals associated with root systems of rank two, J. Math. Anal. Appl. 216 (1997), pp. 1-99. MR 99e:33023
[21]
K.W.J. Kadell, A proof of Askey's conjectured q-analogue of Selberg's integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), pp. 1-99. MR 89h:33006b
[22]
J. Kaneko, $q$-Selberg integrals and Macdonald polynomials, Ann. scient. Éc. Norm. Sup., $4^e$ série, 29 (1996), pp. 583-637. MR 98k:33026
[23]
R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 94-05 (Delft University of Technology, 1994).
[24]
T.H. Koornwinder, Askey-Wilson polynomials for root systems of type BC, Contemp. Math. 138 (1992), pp. 1-99. MR 94e:33039
[25]
I.G. Macdonald, Orthogonal polynomials associated with root systems, preprint (1988).
[26]
I.G. Macdonald, Some conjectures for Koornwinder's orthogonal polynomials, unpublished manuscript (1988).
[27]
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 2nd edition (1995). MR 96h:05027
[28]
I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki No.1 797 (1995). MR 99f:33024
[29]
M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), pp. 1-99. MR 98a:33004
[30]
M. Noumi, Macdonald-Koornwinder polynomials and affine Hecke algebras, RIMS Kokyuroku 919 (1995), pp. 1-99 (in Japanese).
[31]
M. Noumi, T. Sugitani, Quantum symmetric spaces and related $q$-orthogonal polynomials, in 'Group Theoretical Methods in Physics', Proceedings XX ICGTMP, Toyonaka, Japan (eds. A. Arima et al.), World Scientific, Singapore, pp. 28-40. MR 97h:33033
[32]
S. Sahi, Nonsymmetric Koornwinder polynomials and duality, preprint (1997).
[33]
J.V. Stokman, Multivariable big and little q-Jacobi polynomials, Siam J. Math. Anal. 28 (1997), pp. 1-99. MR 98f:33034
[34]
J.V. Stokman, Two limit transitions involving multivariable BC type Askey-Wilson polynomials in 'Quantum Groups and Quantum Spaces' (eds. R. Budzy\'{n}ski, W. Pusz and S. Zakrzewski), Banach Center Publications 40 (1997), pp. 1-99. MR 98m:33046
[35]
J.V. Stokman, Multivariable BC type Askey-Wilson polynomials with partly discrete orthogonality measure, Ramanujan J. 1 (1997), pp. 1-99. CMP 98:08
[36]
J.V. Stokman, T.H. Koornwinder, Limit transitions for BC type multivariable orthogonal polynomials, Canad. J. Math. 49 (1997), pp. 373-404. MR 98g:33031
[37]
J.V. Stokman, T.H. Koornwinder, On some limit cases of Askey-Wilson polynomials, J. Approx. Theory 95 (1998), pp. 1-99. CMP 99:04
[38]
V. Tarasov, A. Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque No. 246 (1997). CMP 99:02

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 33D52, 33D45, 33D80

Retrieve articles in all Journals with MSC (2000): 33D52, 33D45, 33D80

Additional Information:

Jasper V. Stokman
Affiliation: KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Address at time of publication: Centre de Mathématiques de Jussieu, Université Paris 6 Pierre et Marie Curie, 4 Place Jussieu, Paris 75252 Cedex 05, France
Email: stokman@math.jussieu.fr

DOI: 10.1090/S0002-9947-99-02551-9
PII: S 0002-9947(99)02551-9
Keywords: Multivariable basic hypergeometric orthogonal polynomials, Koornwinder polynomials, multivariable $q$-Racah polynomials, multivariable big and little $q$-Jacobi polynomials, $q$-Selberg type integrals, residue calculus
Received by editor(s): July 7, 1997
Posted: November 17, 1999
Additional Notes: The author was supported by a NISSAN-fellowship of the Netherlands Organization of Scientific Research (NWO)
Copyright of article: Copyright 2000, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia