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[1][2]. It is given by

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

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.

See also

• elliptic hypergeometric series
• Jacobi theta function
• Ramanujan theta function

References

1. Gasper, G., Rahman, M. (2004). Basic hypergeometric series. Cambridge university press.
2. Spiridonov, V. P. (2008). Essays on the theory of elliptic hypergeometric functions. Russian Mathematical Surveys, 63(3), 405.