# Basic hypergeometric series psi

The bilateral basic hypergeometric series $\psi$ is defined by $${}_j\psi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_k;q,z)=\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(a_1;q)_k\ldots(a_j;q)_k}{(b_1;q)_k\ldots(b_{\ell};q)_k}\left( (-1)^k q^{ {k \choose 2} } \right)^{\ell-j}z^k.$$
Theorem: The following formula holds: $${}_1\psi_1(a,b;q,z) = \dfrac{\left(\frac{b}{a};q \right)_{\infty} (q,q)_{\infty} \left( \frac{q}{az};q \right)_{\infty} (az;q)_{\infty} }{(b;q)_{\infty} \left( \frac{b}{az};q \right)_{\infty} \left( \frac{q}{a};q \right)_{\infty} (z;q)_{\infty}}.$$