An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums

We present an elliptic Macdonald-Morris constant term conjecture in the for m of an evaluation formula for a Selberg-type multiple beta integral compos ed of elliptic gamma functions. By multivariate residue calculus, a summati on formula recently conjectured by Warnaar for a multiple modular (or ellip tic) hypergeometric series is recovered. When the imaginary part of the mod ular 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 hypergeomet ric degeneration of Warnaar's sum, which amounts to a multidimensional gene ralization of Jackson's very-well-poised balanced terminating (8)Phi (7) su mmation formula. By exploiting its modular properties, the validity of Warn aar's sum at the elliptic level is moreover verified independently for low orders in log(q) (viz. up to order 10).