Solution of functional equations related to elliptic functions

Functional equations of the form $f(x+y) g(x-y) = \sum _{j=1}^n \alpha _j(x)\beta _j(y)$ as well as of the form $f_1(x+z) f_2(y+z) f_3(x+y-z) = \sum _{j=1}^{m} \phi _j(x,y) \psi _j(z)$ are solved for unknown entire functions $f,g,\alpha _j,\beta _j: \mathbb{C} \to \mathbb{C} $ and $f_1,f_2,f_3,\psi _j: \mathbb{C} \to \mathbb{C} $, $\phi _j: \mathbb{C} ^2\to \mathbb{C} $ in the cases of $n=3$ and $m=4$.