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


SIGMA 5 (2009), 055, 20 pages      arXiv:0905.4033      https://doi.org/10.3842/SIGMA.2009.055
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Theta Functions, Elliptic Hypergeometric Series, and Kawanaka's Macdonald Polynomial Conjecture

Robin Langer a, Michael J. Schlosser b and S. Ole Warnaar c
a) Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia
b) Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Vienna, Austria
c) School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

Received March 01, 2009, in final form May 19, 2009; Published online May 25, 2009

Abstract
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.

Key words: theta functions; Macdonald polynomials; elliptic hypergeometric series.

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References

  1. Bhatnagar G., Schlosser M.J., Cn and Dn very-well-poised 10φ9 transformations, Constr. Approx. 14 (1998), 531-567.
  2. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, 171-204.
  3. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  4. Gustafson R.A., Multilateral summation theorems for ordinary and basic hypergeometric series in U(n), SIAM J. Math. Anal. 18 (1987), 1576-1596.
  5. Kajihara Y., Euler transformation formula for multiple basic hypergeometric series of type A and some applications, Adv. Math. 187 (2004), 53-97.
  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. Kawanaka N., On subfield symmetric spaces over a finite field, Osaka J. Math. 28 (1991), 759-791.
  8. Kawanaka N., A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999), 157-176.
  9. Lascoux A., Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, Vol. 99, American Mathematical Society, Providence, RI, 2003.
  10. Lassalle M., Schlosser M.J., Inversion of the Pieri formula for Macdonald polynomials, Adv. Math. 202 (2006), 289-325, math.CO/0402127.
  11. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995.
  12. Milne S.C., Multiple q-series and U(n) generalizations of Ramanujan's 1ψ1 sum, in Ramanujan Revisited (Urbana-Champaign, Ill., 1987), Academic Press, Boston, MA, 1988, 473-524.
  13. Rosengren H., A proof of a multivariable elliptic summation formula conjectured by Warnaar, in q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), Contemp. Math., Vol. 291, Amer. Math. Soc., Providence, RI, 2001, 193-202, math.CA/0101073.
  14. Rosengren H., Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417-447, math.CA/0207046.
  15. Rosengren H., Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux, J. Combin. Theory Ser. A 115 (2008), 376-406, math.CO/0603086.
  16. Rosengren H., Schlosser M.J., On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series, J. Comput. Appl. Math. 178 (2005), 377-391, math.CA/0309358.
  17. Ruijsenaars S.N.M., Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), 191-213.
  18. Ruijsenaars S.N.M., Elliptic integrable systems of Calogero-Moser type: some new results on joint eigenfunctions, in Proceedings of the 2004 Kyoto Workshop on Elliptic Integrable Systems, Editors M. Noumi and K. Takasaki, Rokko Lectures in Mathematics, no. 18, Kobe University, 2005, 223-240.
  19. Ruijsenaars S.N.M., Schneider H., A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370-405.
  20. Schlosser M.J., Multidimensional matrix inversions and Ar and Dr basic hypergeometric series, Ramanujan J. 1 (1997), 243-274.
  21. Schlosser M.J., Explicit computation of the q,t-Littlewood-Richardson coefficients, in Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, Amer. Math. Soc., Providence, RI, 2006, 335-343.
  22. Warnaar S.O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502, math.QA/0001006.
  23. Warnaar S.O., Rogers-Szegö polynomials and Hall-Littlewood symmetric functions, J. Algebra 303 (2006), 810-830, arXiv:0708.3110.
  24. Whittaker E.T., Watson G.N., A course of modern analysis, 4th ed., Cambridge University Press, Cambridge, 1927.


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