Q-theta function

In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. ¹² It is given by

${\displaystyle \theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)}$

where one takes 0 ≤ |q| < 1. It obeys the identities

${\displaystyle \theta (z;q)=\theta \left({\frac {q}{z}};q\right)=-z\theta \left({\frac {1}{z}};q\right).}$

It may also be expressed as:

${\displaystyle \theta (z;q)=(z;q)_{\infty }(q/z;q)_{\infty }}$

where ${\displaystyle (\cdot \cdot )_{\infty }}$ is the q-Pochhammer symbol.