https://en.formulasearchengine.com/api.php?action=feedcontributions&feedformat=atom&user=143.107.45.30formulasearchengine - User contributions [en]2026-05-02T03:35:19ZUser contributionsMediaWiki 1.43.0-wmf.28https://en.formulasearchengine.com/index.php?title=Pumping_lemma_for_regular_languages&diff=9938Pumping lemma for regular languages2013-10-15T11:48:17Z<p>143.107.45.30: /* Formal statement */</p>
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{{unreferenced|date=August 2012}}<br />
In [[mathematics]], the '''q-theta function''' is a type of [[q-series]]. It is given by<br />
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:<math>\theta(z;q)=\prod_{n=0}^\infty (1-q^nz)\left(1-q^{n+1}/z\right)</math><br />
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where one takes 0&nbsp;&le;&nbsp;|''q''|&nbsp;<&nbsp;1. It obeys the identities<br />
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:<math>\theta(z;q)=\theta\left(\frac{q}{z};q\right)=-z\theta\left(\frac{1}{z};q\right). </math><br />
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It may also be expressed as:<br />
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:<math>\theta(z;q)=(z;q)_\infty (q/z;q)_\infty</math><br />
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where <math>(\cdot \cdot )_\infty</math> is the [[q-Pochhammer symbol]].<br />
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==See also==<br />
* [[theta function|Jacobi theta function]]<br />
* [[Ramanujan theta function]]<br />
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[[Category:Q-analogs]]<br />
[[Category:Theta functions]]<br />
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{{combin-stub}}</div>143.107.45.30