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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