Far Eastern Mathematical Journal

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Solution of functional equations related to elliptic functions. III


Illarionov A.A., Markova N.V.

2019, issue 2, Ń. 197–205


Abstract
Let $s, m\in \Bbb N$, $s\ge2$. We solve the functional equation $$f_1(x_1+z)\ldots f_{s-1}(x_{s-1}+z)f_s(x_1+\ldots+x_{s-1}-z)=\sum_{j=1}^{m}\varphi_j(x_1,\ldots,x_{s-1})\psi_j(z),$$ for unknown entire functions $f_1,\ldots,f_s:\Bbb C\to\Bbb C$, $\varphi_j:\Bbb C^{s-1}\to \Bbb C$, $\psi_j:\Bbb C\to\Bbb C$ in the case of $s\ge3$, $m\le2s-1$. All non-elementary solutions are described by the Weierstrass sigma-function. Previously, such results were known for $m\le s+1$. The considered equation arises in the study of polylinear functional-differential operators and multidimensional vector addition theorems.

Keywords:
addition theorem, functional equation, Weierstrass sigma-function, theta function, elliptic function

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