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Hyperquasipolynomials and their applications. (English. Russian original) Zbl 1360.30023

Funct. Anal. Appl. 50, No. 3, 193-203 (2016); translation from Funkts. Anal. Prilozh. 50, No. 3, 34-46 (2016).
Summary: For a given nonzero entire function \(g: \mathbb{C}\to\mathbb{C}\), we study the linear space \(F(g)\) of all entire functions \(f\) such that \[ f\left( {z + w} \right)g\left( {z - w} \right) = {\phi _1}\left( z \right){\psi _1}\left( w \right) + \cdots + \phi_n \left( z \right){\psi _n}\left( w \right), \] where \(\phi_{1},\psi_{1},\dots,\phi_n,\psi_n: \mathbb{C}\to\mathbb{C}\). 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.

MSC:

30D20 Entire functions of one complex variable (general theory)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B32 Functional equations for complex functions
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