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1
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Coefficients of powers of the theta function
Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$
Now, I shall show that the powers of $\theta$ are given by
$$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$
where $S_r(n)$ ...
5
votes
1answer
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Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?
In my recent explorations I stumbled upon the following series
$$
\vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...
