Ramanujan's identity

Statement

Srinivasa Ramanujan gave the identity

$$\;_1\psi_1 \left[\begin{matrix} a \\ b \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a;q)_n} {(b;q)_n} z^n = \frac {(b/a,q,q/az,az;q)_\infty } {(b,b/az,q/a,z;q)_\infty}$$

valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for $\;_6\psi_6$ have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as

$$\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n = (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.$$

Gwynneth Coogan and Ken Ono give a related formal power series

$$A(z;q) \stackrel{\rm{def}}{=} \frac{1}{1+z} \sum_{n=0}^\infty \frac{(z;q)_n}{(-zq;q)_n}z^n = \sum_{n=0}^\infty (-1)^n z^{2n} q^{n^2}.$$