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An elliptic beta integral. (English) Zbl 1065.33015

Elaydi, S. (ed.) et al., New trends in difference equations. Proceedings of the 5th international conference on difference equations and applications, Temuco, Chile, January 2–7, 2000. London: Taylor & Francis (ISBN 0-415-28389-2/hbk). 273-282 (2002).
The authors propose a new type of integral representing an elliptic generalization of a \(q\)-beta integral. It depends on five complex parameters \(t_m\), \(m= 0,\dots, 4\), and two complex bases \(p\) and \(q\). The corresponding formula expresses a contour integral over \(z\in\mathbb C\) of a ratio of some elliptic gamma functions depending on \(z\) and \(t_m\) as another ratio of these gamma functions up to some multiplicative factor depending only on \(p\) and \(q\). This conjectured formula is proved for a particular set of values of the parameters \(t_m\), when one of the parameters is continuous and four others take on a finite number of discrete values.
For the entire collection see [Zbl 1050.39001].

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
33E05 Elliptic functions and integrals
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