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. 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.

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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

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.

See also

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

References

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1. ^ Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. doi:10.1017/CBO9780511526251. ISBN 9780521833578.
2. ^ Spiridonov, V. P. (2008). "Essays on the theory of elliptic hypergeometric functions". Russian Mathematical Surveys. 63 (3): 405–472. arXiv:0805.3135. Bibcode:2008RuMaS..63..405S. doi:10.1070/RM2008v063n03ABEH004533. S2CID 16996893.

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