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


SIGMA 3 (2007), 106, 9 pages      arXiv:0711.2401      https://doi.org/10.3842/SIGMA.2007.106
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Biorthogonal Expansion of Non-Symmetric Jack Functions

Siddhartha Sahi a and Genkai Zhang b
a) Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA
b) Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Göteborg University, Sweden

Received August 08, 2007, in final form October 31, 2007; Published online November 15, 2007

Abstract
We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.

Key words: non-symmetric Jack polynomials and functions; biorthogonal expansion; Laplace transform; Cherednik-Opdam transform.

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References

  1. Baker T.H., Forrester P.J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), 175-216, solv-int/9608004.
  2. Baker T.H., Forrester P.J., Non-symmetric Jack polynomials and intergral kernels, Duke Math. J. 95 (1998), 1-50, q-alg/9612003.
  3. Davidson M., Olafsson G., Zhang G., Segal-Bargmann transform on Hermitian symmetric spaces, J. Funct. Anal. 204 (2003), 157-195, math.RT/0206275.
  4. Faraut J., Koranyi A., Analysis on symmetric cones, Oxford University Press, Oxford, 1994.
  5. Knop F., Sahi S., A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 9-22, q-alg/9610016.
  6. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Math. Report 98-17, Delft Univ. of Technology, 1998, math.CA/9602214.
  7. Macdonald I.G., Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1995.
  8. Macdonald I.G., Hypergeometric functions, Lecture notes, unpublished.
  9. Peng L., Zhang G., Nonsymmetric Jacobi and Wilson-type polynomials, Int. Math. Res. Not. (2006), Art. ID. 21630, 13 pages, math.CA/0511709.
  10. Opdam E.M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121.
  11. Sahi S., A new scalar product for nonsymmetric Jack polynomials, Int. Math. Res. Not. (1996), no. 20, 997-1004, q-alg/9608013.
  12. Sahi S., The binomial formula for nonsymmetric Macdonald polynomials, Duke Math. J. 94 (1998), 465-477, q-alg/9703024.
  13. Sahi S., The spectrum of certain invariant differential operators associated to a Hermitian symmetric space, in Lie Theory and Geometry, Progr. Math., Vol. 123, Birkhäuser, Boston MA, 1994, 569-576.
  14. Zhang G., Branching coefficients of holomorphic representations and Segal-Bargmann transform, J. Funct. Anal. 195 (2002), 306-349, math.RT/0110212.
  15. Zhang G., Spherical transform and Jacobi polynomials on root systems of type BC, Int. Math. Res. Not. (2005), no. 51, 3169-3190, math.RT/0503735.


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