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On integral representations of \(q\)-gamma and \(q\)-beta functions. (English) Zbl 1225.33017
Summary: We study \(q\)-integral representations of the \(q\)-gamma and the \(q\)-beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi’s identity, and of Ramanujan’s formula for the bilateral hypergeometric series.

MSC:
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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