Hyperquasipolynomials and their applications

For a given nonzero entire function $g\colon\mathbb{C}\to\mathbb{C}$, we study the linear space $\mathcal{F}(g)$ of all entire functions $f$ such that
$$f(z+w)g(z-w)=\varphi_1(z)\psi_1(w)+\dots+\varphi_n(z)\psi_n(w),$$
where $\varphi_1, \psi_1, \dots,\varphi_n,\psi_n\colon\mathbb{C}\to\mathbb{C}$. In the case of $g\equiv1$, the expansion characterizes quasipolynomials, that is, linear combinations of products of polynomials by exponential functions. (This is a theorem due to Levi-Civita.) As an application, all solutions of a functional equation in the theory of trilinear functional equations are obtained.