Geodesic
Functional Equations and Weierstrass Sigma-Functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 4, pp. 43-54.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that if an entire function $f\colon\mathbb{C}\to\mathbb{C}$ satisfies an equation of the form $f(x+y) f(x-y) = \alpha_1(x)\beta_1(y)+ \alpha_2(x)\beta_2(y) + \alpha_3(x)\beta_3(y)$, $x,y\in \mathbb{C}$, for some $\alpha_j,\beta_j\colon\mathbb{C}\to\mathbb{C}$ and there exist no $\tilde \alpha_j$ and $\tilde\beta_j$ for which $f(x+y) f(x-y) = \tilde\alpha_1(x)\tilde\beta_1(y)+ \tilde\alpha_2(x)\tilde\beta_2(y)$, then $f(z) = \exp(Az^2+ Bz + C) \cdot \sigma_\Gamma (z-z_1)\cdot \sigma_\Gamma (z-z_2)$, where $\Gamma$ is a lattice in $\mathbb{C}$; $\sigma_\Gamma$ is the Weierstrass sigma-function associated with $\Gamma$; $A,B,C,z_1,z_2\in\mathbb{C}$; and $z_1-z_2\notin (\frac{1}{2}\Gamma)\setminus \Gamma$.
Keywords: functional equation, Weierstrass sigma-function, elliptic function, addition theorem, trilinear functional equation. 
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A. A. Illarionov. Functional Equations and Weierstrass Sigma-Functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 4, pp. 43-54. http://geodesic.mathdoc.fr/item/FAA_2016_50_4_a4/

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