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.)