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Some quadratic and cubic summation formulas for basic hypergeometric series. (English) Zbl 0774.33012

The author obtains five elegant summation formulas for basic hypergeometric series with bases \(q\) and \(q^ 2\); \(q\) and \(q^ 3\) where \(0<q<1\). The proofs make use of the \(q\)-binomial theorem, the \(q\)-Gauss formula,the \(q\)-Saalschütz formula and an identity of Carlitz. A few special cases of the formulas provide interesting known results.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D65 Bibasic functions and multiple bases
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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