Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 002, 11 pages      math.QA/0701134
Contribution to the Vadim Kuznetsov Memorial Issue

Raising and Lowering Operators for Askey-Wilson Polynomials

Siddhartha Sahi
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

Received September 20, 2006, in final form December 27, 2006; Published online January 04, 2007

In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.

Key words: orthogonal polynomials; Askey-Wilson polynomials; q-difference equation; three term recurrence; raising operators; lowering operators; root systems; double affine Hecke algebra.

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  1. Askey R., Wilson J., Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319 (1985), 1-53.
  2. Bangerezako G., The factorization method for the Askey-Wilson polynomials, J. Comput. Appl. Math. 107 (1999), 219-232.
  3. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. (1992), no. 9, 171-180.
  4. Cherednik I., Double affine Hecke algebras and Macdonald's conjectures, Ann. of Math. 141 (1995), 191-216.
  5. Etingof P., Oblomkov A., Rains E., Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces, math.QA/0406480.
  6. Garsia A., Remmel R., Plethystic formulas and positivity for q,t-Kostka coefficients, in Mathematical Essays in Honor of Gian-Carlo Rota, Editors B. Sagan and R. Stanley, Progr. Math. 161 (1998), 245-262.
  7. Garsia A., Tesler G., Plethystic formulas for the Macdonald q,t-Kostka coefficients, Adv. Math. 123 (1996), 144-222.
  8. Ion B., Sahi S., Triple groups and Cherednik algebras, Contemp. Math. 417 (2006), 183-206, math.QA/0304186.
  9. Kirillov A., Noumi M., q-difference raising operators for Macdonald polynomials and the integrality of transition coefficients, in Algebraic Methods and q-Special Functions, CRM Proceedings and Lecture Notes 22 (1999), 227-243, q-alg/9605005.
  10. Kirillov A., Noumi M., Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998), 1-39, q-alg/9605004.
  11. Knop F., Integrality of two variable Kostka functions, J. Reine Angew. Math. 482 (1997), 177-189, q-alg/9603027.
  12. Koekoek R., Swarttouw R., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Department of Technical Mathematics and Informatics, Report no. 98-17 (1998),
  13. Koornwinder T., Askey-Wilson polynomials for root systems of type BC, Contemp. Math. 138 (1992), 189-204.
  14. Koornwinder T., Lowering and raising operators for some special orthogonal polynomials, Contemp. Math. 417 (2006), 227-238.
  15. Koornwinder T., The structure relation for Askey-Wilson polynomials, J. Comput. Appl. Math., to appear, math.CA/0601303.
  16. Lapointe L., Vinet L., Creation operators for the Macdonald and Jack polynomials, Lett. Math. Phys. 40 (1997), 269-286.
  17. Lapointe L., Vinet L., Rodrigues formulas for the Macdonald polynomials, Adv. Math. 130 (1997), 261-279, q-alg/9607025.
  18. Macdonald I., Affine Hecke algebras and orthogonal polynomials, Cambridge University Press, 2003.
  19. Noumi M., Macdonald-Koornwinder polynomials and affine Hecke algebras, RIMS Kokyuroku 919 (1995), 44-55 (in Japanese).
  20. Noumi M., Stokman J., Askey-Wilson polynomials: an affine Hecke algebra approach, in Laredo Lectures on Orthogonal Polynomials and Special Functions, Editors R. Alvarez-Nodarse, F. Marcellan and W. Van Assche, Nova Science Publishers, 2004, 111-144, math.QA/0001033.
  21. Sahi S., Interpolation, integrality, and a generalization of Macdonald's polynomials, Int. Math. Res. Not. (1996), no. 10, 457-471.
  22. Sahi S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. 150 (1999), 267-282, q-alg/9710032.
  23. Sahi S., Some properties of Koornwinder polynomials, Contemp. Math. 254 (2000), 395-411.
  24. Stokman J., Koornwinder polynomials and affine Hecke algebras, Int. Math. Res. Not. (2000), no. 19, 1005-1042, math.QA/0002090.
  25. van Diejen J., Self-dual Koornwinder-Macdonald polynomials, Invent. Math. 126 (1996), 319-339, q-alg/9507033.

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