Q-theta function: Difference between revisions

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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

\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

\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.

References

^ Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. doi:10.1017/CBO9780511526251. ISBN 9780521833578.
^ 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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[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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+{{DISPLAYTITLE:''q''-theta function}}
-In [[mathematics]], the '''q-theta function''' is a type of [[q-series]]. It is given by
+{{more citations needed|date=January 2021}}
+In [[mathematics]], the '''''q''-theta function''' (or modified Jacobi theta function) is a type of [[q-series|''q''-series]] which is used to define [[elliptic hypergeometric series]].
+<ref>{{cite book |doi=10.1017/CBO9780511526251|title=Basic Hypergeometric Series |year=2004 |last1=Gasper |first1=George |last2=Rahman |first2=Mizan |isbn=9780521833578 }}</ref><ref>{{cite journal|s2cid=16996893 |doi=10.1070/RM2008v063n03ABEH004533 |title=Essays on the theory of elliptic hypergeometric functions |year=2008 |last1=Spiridonov |first1=V. P. |journal=Russian Mathematical Surveys |volume=63 |issue=3 |pages=405–472 |arxiv=0805.3135 |bibcode=2008RuMaS..63..405S }}</ref> It is given by
-:<math>\theta(z;q)=\prod_{n=0}^\infty (1-q^nz)(1-q^{n+1}/z).</math>
+:<math>\theta(z;q):=\prod_{n=0}^\infty (1-q^nz)\left(1-q^{n+1}/z\right)</math>
-where one takes <math>0\le|q|<1.</math> It obeys the identities
+where one takes 0&nbsp;&le;&nbsp;|''q''|&nbsp;<&nbsp;1.  It obeys the identities
-:<math>\theta(z;q)=\theta(q/z;q)=-z\theta(1/z;q)</math>
+:<math>\theta(z;q)=\theta\left(\frac{q}{z};q\right)=-z\theta\left(\frac{1}{z};q\right). </math>
+It may also be expressed as:
+:<math>\theta(z;q)=(z;q)_\infty (q/z;q)_\infty</math>
+where <math>(\cdot  \cdot )_\infty</math> is the [[q-Pochhammer symbol]].
 ==See also==
+* [[elliptic hypergeometric series]]
 * [[theta function|Jacobi theta function]]
+* [[Ramanujan theta function]]
+==References==
+{{reflist}}
-{{mathstub}}
 [[Category:Q-analogs]]
+[[Category:Theta functions]]
+{{combin-stub}}