Geodesic
Hyperquasipolynomials and their applications
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 3, pp. 34-46

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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.
Mots-clés : quasipolynomial
Keywords: Weierstrass sigma function, trilinear functional equation. 
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     author = {V. A. Bykovskii},
     title = {Hyperquasipolynomials and their applications},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {34--46},
     publisher = {mathdoc},
     volume = {50},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2016_50_3_a2/}
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V. A. Bykovskii. Hyperquasipolynomials and their applications. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 3, pp. 34-46. http://geodesic.mathdoc.fr/item/FAA_2016_50_3_a2/