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Zbl 1014.33011
Van Diejen, J.F.; Spiridonov, V.P.
Modular hypergeometric residue sums of elliptic Selberg integrals. (English)
[J] Lett. Math. Phys. 58, No.3, 223-238 (2001). ISSN 0377-9017; ISSN 1573-0530/e

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 $_8\varepsilon _7$ 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.
[Luigi Gatteschi (Torino)]

MSC 2000:: *33E05 Elliptic functions and integrals
33C67 Hypergeometric functions associated with root systems
11F50 Jacobi forms
11L07 Estimates on exponential sums

Keywords: hypergeometric sums; elliptic Selberg integrals; residue calculus; Jacobi forms

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Zentralblatt MATH Berlin [Germany]