Geodesic
Hyperquasipolynomials for the Theta-Function
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 84-87

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Let $g$ be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions $f\colon\mathbb{C}\to\mathbb{C}$ satisfying $f(x+y)g(x-y)=\alpha_1(x)\beta_1(y)+\cdots+\alpha_r(x)\beta_r(y)$ for some $r\in\mathbb{N}$ and $\alpha_j,\beta_j\colon\mathbb{C}\to\mathbb{C}$ are described.
Keywords: addition theorem, Jacobi theta function, Weierstrass sigma function, elliptic function, functional equation. 
@article{FAA_2018_52_3_a7,
     author = {A. A. Illarionov and M. A. Romanov},
     title = {Hyperquasipolynomials for the {Theta-Function}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {84--87},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a7/}
}
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A. A. Illarionov; M. A. Romanov. Hyperquasipolynomials for the Theta-Function. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 84-87. http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a7/