ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 32, Number 2, Summer 2002 ELLIPTIC BETA INTEGRALS AND MODULAR HYPERGEOMETRIC SUMS: AN OVERVIEW J.F. VAN DIEJEN AND V.P. SPIRIDONOV ABSTRACT. Recent results on elliptic generalizations of various beta integrals are reviewed. Firstly, a single vari- able Askey-Wilson type integral describing an elliptic exten- sion 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 nonre- duced root systems is described. The Frenkel-Turaev sum and its multivariable generalization, conjectured recently by War- naar, 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 gen- eralization of the Denis-Gustafson sum a modular extension of the Milne-type multiple basic hypergeometric sums. 1. Introduction. Elliptic generalizations of the very well-poised ba- sic hypergeometric series were introduced by Frenkel and Turaev [12] in relation to elliptic solutions of the Yang-Baxter equation associated with the SOS-type solvable models of statistical mechanics [5]. These series were derived also in [30] through a different technique, as solu- tions of some particular spectral problems associated with new families of discrete biorthogonal rational functions generalizing the Wilson’s functions [33, 34]. Various nice properties of the elliptic hypergeomet- ric series were discovered in [12]. Firstly, under a balancing condition, they become invariant with respect to modular transformations. Sec- ondly, there exist natural generalizations of the Bailey’s transforma- tion formula for a terminating 10 Φ 9 series and of the Jackson’s sum for a terminating 8 Φ 7 series. Some new identities for terminating ellip- tic hypergeometric series were derived and a multiple extension of the Frenkel-Turaev sum was conjectured by Warnaar in [32]. Work supported in part by the Fondo Nacional de Desarrollo Cient´ ıfico y Tec- nol´ogico (FONDECYT) Grants #1989832 and #7980041, the Programa Formas Cuadr´aticas of the Universidad de Talca, the C´atedra Presidencial in Number The- ory, and by the Russian Foundation for Basic Research (RFBR) grant #1980832. Proceedings of the NSF Conference, Special Functions 2000 (Tempe, USA, May 29 June 9, 2000). Received by the editors on October 31, 2000. Copyright c 2002 Rocky Mountain Mathematics Consortium 639