Functional Equations and Weierstrass Sigma-Functions

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 $z_1-z_2\notin (\frac{1}{2}\Gamma)\setminus \Gamma$.