Abstract:
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.
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