Geodesic
On the rank of a finite set of theta functions
Dalʹnevostočnyj matematičeskij žurnal, Tome 16 (2016) no. 2, pp. 181-185

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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. 
@article{DVMG_2016_16_2_a6,
     author = {M. D. Monina},
     title = {On the rank of a finite set of theta functions},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {181--185},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2016_16_2_a6/}
}
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M. D. Monina. On the rank of a finite set of theta functions. Dalʹnevostočnyj matematičeskij žurnal, Tome 16 (2016) no. 2, pp. 181-185. http://geodesic.mathdoc.fr/item/DVMG_2016_16_2_a6/