Hyperquasipolynomials and their applications

Abstract

For a given nonzero entire function g: C → C, we study the linear space F(g) of all entire functions f such that

$$f\left( {z + w} \right)g\left( {z - w} \right) = {\varphi _1}\left( z \right){\psi _1}\left( w \right) + \cdots \varphi {n_{}}\left( z \right){\psi _n}\left( w \right),$$

where φ ₁, ψ ₁,..., φ _n , ψ _n : C → C. In the case of g ≡ 1, 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.