Abstract

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 +infinity, 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).