Elliptic beta integrals and modular hypergeometric sums: An overview

Abstract

Recent results on elliptic generalizations of various beta integrals are reviewed. Firstly, a single variable Askey-Wilson type integral describing an elliptic extension of the Nassrallah-Rahman integral is presented. Then a multiple Selberg-type integral defining an elliptic extension of the Macdonald-Morris constant term identities for nonreduced root systems is described. The Frenkel-Turaevsu m and its multivariable generalization, conjectured recently by Warnaar, follow from these integrals through residue calculus. A new elliptic Selberg-type integral, from which the previous one can be derived via a technique due to Gustafson, is defined. Residue calculus applied to this integral yields an elliptic generalization of the Denis-Gustafson sum a modular extension of the Milne-type multiple basic hypergeometric sums.