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. It is given by

θ ( z ; q ) := ∏ n = 0 ∞ ( 1 − q n z ) ( 1 − q n + 1 / z ) {\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

θ ( z ; q ) = θ ( q z ; q ) = − z θ ( 1 z ; q ) . {\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:

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

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