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In mathematics , the q -theta functionq -serieselliptic hypergeometric series .[1] [2]
θ
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∏
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=
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∞
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q
n
z
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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
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q
)
=
θ
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q
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θ
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.
{\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
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z
;
q
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=
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z
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q
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∞
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z
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q
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∞
{\displaystyle \theta (z;q)=(z;q)_{\infty }(q/z;q)_{\infty }}
where
(
⋅
⋅
)
∞
{\displaystyle (\cdot \cdot )_{\infty }}
q-Pochhammer symbol .
See also [ ] References [ ]
^ Gasper, G., Rahman, M. (2004). Basic hypergeometric series. Cambridge university press.
^ Spiridonov, V. P. (2008). Essays on the theory of elliptic hypergeometric functions. Russian Mathematical Surveys, 63(3), 405.
