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


SIGMA 5 (2009), 054, 40 pages      arXiv:0812.0279      https://doi.org/10.3842/SIGMA.2009.054
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications

Yasushi Komori a, Masatoshi Noumi b and Jun'ichi Shiraishi c
a) Graduate School of Mathematics, Nagoya University, Chikusa-Ku, Nagoya 464-8602, Japan
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
c) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received December 01, 2008, in final form April 30, 2009; Published online May 12, 2009

Abstract
A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.

Key words: kernel function; Ruijsenaars operator; Koornwinder polynomial.

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References

  1. Aomoto K., Ito M., A determinant formula for a holonomic q-difference system associated with Jackson integrals of type BCn, Adv. Math., to appear.
  2. van Diejen J.F., Integrability of difference Calogero-Moser systems, J. Math. Phys. 35 (1994), 2983-3004.
  3. van Diejen J.F., Self-dual Koornwinder-Macdonald polynomials, Invent. Math. 126 (1996), 319-339, q-alg/9507033.
  4. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  5. Kajihara Y., Noumi M., Raising operators of row type for Macdonald polynomials, Compositio Math. 120 (2000), 119-136, math.QA/9803151.
  6. Kajihara Y., Noumi M., Multiple elliptic hypergeometric series. An approach from the Cauchy determinant, Indag. Math. (N.S.) 14 (2003), 395-421, math.CA/0306219.
  7. Kirillov A.N., Noumi M., Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998), 1-39, q-alg/9605004.
  8. Kirillov A.N., Noumi M., q-difference raising operators for Macdonald polynomials and the integrality of transition coefficients, in Algebraic Methods and q-Special Functions (Montréal, QC, 1996), CRM Proc. Lecture Notes, Vol. 22, Amer. Math. Soc., Providence, RI, 1999, 227-243, q-alg/9605005.
  9. Komori Y., Hikami K., Quantum integrability of the generalized elliptic Ruijsenaars models, J. Phys. A: Math. Gen. 30 (1997), 4341-4364.
  10. Koornwinder T.H., Askey-Wilson polynomials for root systems of type BC, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204.
  11. Lassalle M., Some conjectures for Macdonald polynomials of type B, C, D, Sém. Lothar. Combin. 52 (2004), Art. B52h, 24 pages, math.CO/0503149.
  12. Langmann E., An explicit solution of the (quantum) elliptic Calogero-Sutherland model, in SPT 2004-Symmetry and Perturbation Theory, World Sci. Publ., Hackensack, NJ, 2005, 159-174, math-ph/0407050.
  13. Langmann E., Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model, J. Math. Phys. 47 (2006), 022101, 18 pages, math-ph/0406061.
  14. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995.
  15. Mimachi K., A duality of Macdonald-Koornwinder polynomials and its applications to integral representations, Duke Math. J. 107 (2001), 263-281.
  16. Mimachi K., Noumi M., An integral representation of eigenfunctions for Macdonald's q-difference operators, Tôhoku Math. J. 49 (1997), 517-525.
  17. Mimachi K., Noumi M., A reproducing kernel for nonsymmetric Macdonald polynomials, Duke Math. J. 91 (1998), 621-634, q-alg/9610014.
  18. Okounkov A., BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), 181-207, q-alg/9611011.
  19. Rains E.M., BCn-symmetric polynomials, Transform. Groups 10 (2005), 63-132, math.QA/0112035.
  20. Ruijsenaars S.N.M., Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), 191-213.
  21. Ruijsenaars S.N.M., Elliptic integrable systems of Calogero-Moser type: some new results on joint eigenfunctions, in Elliptic Integrable Systems, Proceedings of the Workshop on Elliptic Integrable Systems (Kyoto, 2004), Rokko Lectures in Mathematics, Vol. 18, Kobe University, 2005, 223-240.
  22. Ruijsenaars S.N.M., Eigenfunctions with a zero eigenvalue for differences of elliptic relativistic Calogero-Moser Hamiltonians, Theoret. and Math. Phys. 146 (2006), 25-33.
  23. Ruijsenaars S.N.M., Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type. I. The eigenfunction identities, Comm. Math. Phys. 286 (2009), 629-657.
    Ruijsenaars S.N.M., Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type. II. The AN – 1 case: first steps, Comm. Math. Phys. 286 (2009), 659-680.
  24. Stokman J.V., Lecture notes on Koornwinder polynomials, in Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004, 145-207.

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