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An interesting generalization of Jacobi’s triple product identity. (English) Zbl 1049.33011

The “interesting” identity in question is \[ \sum_{m=0}^{\infty} \frac{q^{m(m+1)/2}}{(cz)_{m+1}}z^m + \sum_{m=1}^{\infty} \frac{q^{m(m-1)/2}}{(-c)_{m}}z^{-m} = \frac{(-qz)_{\infty}(-1/z)_{\infty}(q)_{\infty}}{(-c)_{\infty}(cz)_{\infty}}, \] where \[ (a)_n = \prod_{k=0}^{n-1} (1-aq^k). \] This identity follows immediately from an application of the second iteration of Heine’s transformation to a special case of Ramanujan’s \(_1 \psi _1\) summation.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11F20 Dedekind eta function, Dedekind sums
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