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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.
[1]
[2] It is given by
![{\displaystyle \theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9bb1d505ff7295761ba01844fc185d1821a2d9)
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0468b0f1dc93c5958494fde3b12015401542dc)
It may also be expressed as:
![{\displaystyle \theta (z;q)=(z;q)_{\infty }(q/z;q)_{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1481a013ec5cac56dae95cdc90855b1ef017e05a)
where
is the
q-Pochhammer symbol.
See also
References