Geodesic
On a Multilinear Functional Equation
Matematičeskie zametki, Tome 107 (2020) no. 1, pp. 59-73

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The following functional equation is solved: $$ f(x_1+z)\dotsb f(x_2+z)f(x_1+\dotsb+x_{s-1}-z) =\phi_1(x)\psi_1(z)+\dotsb+\phi_m(x)\psi_m(z), $$ where $x=(x_1,\dots,x_{s-1})$, for the unknowns $f,\psi_j\colon\mathbb C\to\mathbb C$ and $\phi_j\colon\mathbb C^{s-1}\to\mathbb C$ for $s\ge 3$ and $m\le 4s-5$.
Keywords: functional equation, theta function, Weierstrass sigma function, elliptic function, addition theorems, multilinear functional-differential operators. 
@article{MZM_2020_107_1_a5,
     author = {A. A. Illarionov},
     title = {On a {Multilinear} {Functional} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {59--73},
     publisher = {mathdoc},
     volume = {107},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_1_a5/}
}
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A. A. Illarionov. On a Multilinear Functional Equation. Matematičeskie zametki, Tome 107 (2020) no. 1, pp. 59-73. http://geodesic.mathdoc.fr/item/MZM_2020_107_1_a5/