q-theta function

In mathematics, the q-theta function is a type of q-series. It is given by

$\theta(z;q)=\prod_{n=0}^\infty (1-q^nz)\left(1-q^{n+1}/z\right)$

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

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

It may also be expressed as:

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

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

See also

• Jacobi theta function
• Ramanujan theta function