On the rank of a finite set of theta functions

In the work for the theta function
$$ \theta(z)=\theta(z;q)=\sum\limits_{n=-\infty}^{\infty}e^{2izn}q^{n^2} $$
identity
\begin{gather*} \theta(z_1+w)\dots\theta(z_{k-1}+w)\theta(z_1+\dots+z_{k-1}-w)=\sum\limits_{i=1}^{s}\varphi_i(z_1,\dots,z_{k-1})\psi_i(w) \\ (\forall z_1,\dots, z_{k-1}, w \in \mathbb{C}) \end{gather*}
with some clearly indicates theta functions $\psi_i$ of one variable and functions $\varphi_i$ of $k-1$ variables is proved.