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Modular hypergeometric residue sums of elliptic Selberg integrals. (English) Zbl 1014.33011
Authors’ abstract: It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the integral then implies a closed summation formula for the series, generalizing both the multiple basic hypergeometric ${}_{8}{{\Phi }}_{7}$ sums of Milne-Gustafson type and the (one-dimensional) modular hypergeometric ${}_{8}{\epsilon }_{7}$ sum of Frenkel and Turaev. Independently, the modular invariance ensures the asymptotic correctness of our multiple modular hypergeometric summation formula for low orders in a modular parameter.
##### MSC:
 33E05 Elliptic functions and integrals 33C67 Hypergeometric functions associated with root systems 11F50 Jacobi forms 11L07 Estimates on exponential sums