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In my recent explorations I stumbled upon the following series

$$ \vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z \in \mathbb{C} $$

This is one of the well known Jacobi theta functions/series with the peculiarity of having the variable $z \in \mathbb{C}$ in a different place, i.e. $e^{\alpha \cdot \mathbf{z} \cdot k^{2}}$!!

The usual form of the theta function is

\begin{align*} \vartheta_{4}(z,e^{\alpha })=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot k^{2}}\cos(2kz) ; \end{align*}

but not in the case I have in hands. Does the former formula make any sense? Where are this kind of series used or analysed? (Apart from the well known case of

$$\psi(x)=\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2} \left[ \vartheta_{3}(0,e^{-\pi x})-1 \right]$$ used in the context of the Riemann zeta-function.)

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For convergence we need $\mathrm{Im} (\alpha\cdot z) < 0$, right? –  GEdgar Aug 6 '11 at 13:27
    
@GEdgar, thats right. –  Neves Aug 6 '11 at 14:38
    
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1 Answer

I've seen this series in Ono's book "The web of modularity" in Theorem 1.60 on page 17 where he gives the identity: $$\frac{\eta(z)^2}{\eta(2z)} = \sum_{n= -\infty}^\infty (-1)^n q^{n^2}$$ where $q = e^{\pi i z}$ and

$$\eta(z) := q^{1/24}\prod_{n=1}^\infty (1-q^n).$$ is the Dedekind Eta function.

I suppose one reason that this kind of thing could be useful is that it is very sparse, and therefore fast to compute lots of coefficients very quickly. For example you could use it to calculate the $q$-expansion of the Eisenstein series $E_4$ using the identity (equation 1.28 in Ono) $$E_4 = \frac{\eta(z)^{16}}{\eta(2z)^8} + 2^8 \frac{\eta(2z)^{16}}{\eta(z)^8}.$$

(However I think there are faster ways to calculate $E_4$?!)

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