Mathematical Research Letters

Volume 7 (2000)

Number 6

An Elliptic Macdonald-Morris Conjecture and Multiple Modular Hypergeometric Sums

Pages: 729 – 746



J. F. van Diejen (Universidad de Chile)

V. P. Spiridonov (Joint Institute for Nuclear Research)


We present an elliptic Macdonald-Morris constant term conjecture in the form of an evaluation formula for a Selberg-type multiple beta integral composed of elliptic gamma functions. By multivariate residue calculus, a summation formula recently conjectured by Warnaar for a multiple modular (or elliptic) hypergeometric series is recovered. When the imaginary part of the modular parameter tends to $+\infty$, our elliptic Macdonald-Morris conjecture follows from a Selberg-type multivariate Nassrallah-Rahman integral due to Gustafson. As a consequence we arrive at a proof for the basic hypergeometric degeneration of Warnaar’s sum, which amounts to a multidimensional generalization of Jackson’s very-well-poised balanced terminating ${}_8\Phi_7$ summation formula. By exploiting its modular properties, the validity of Warnaar’s sum at the elliptic level is moreover verified independently for low orders in $\log (q)$ (viz. up to order $10$).

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