Hyperquasipolynomials for the Theta-Function

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.