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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

Mathematics Encyclopedia

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