## 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

### Keywords:

Jacobi triple product; eta function