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Sib. Èlektron. Mat. Izv., 2019, Volume 16, Pages 481–492 (Mi semr1072)  

Real, complex and functional analysis

Solution of functional equations related to elliptic functions. II

A. A. Illarionovab

a Khabarovsk Division of the Institute of Applied Mathematics, Far Eastern Branch of the Russian Academy of Sciences, 54, Dzerzhinsky str., Khabarovsk, 680000, Russia
b Pacific National University, 136, Tihookeanskaya str., Khabarovsk, 680035, Russia

Abstract: Let $s,m, d\in \mathbb{N}$, $s\ge 2$. We solve the functional equation
\begin{gather*} f_1(\mathbf{u}_1+\mathbf{v})\ldots f_{s-1}(\mathbf{u}_{s-1}+\mathbf{v})f_s(\mathbf{u}_1+\ldots +\mathbf{u}_{s-1}-\mathbf{v}) =\sum_{j=1}^{m} \phi_j(\mathbf{u}_1,\ldots,\mathbf{u}_{s-1})\psi_j(\mathbf{v}), \end{gather*}
for unknown entire functions $f_1,\ldots,f_s:\mathbb{C}^d\to \mathbb{C}$, $\phi_j: (\mathbb{C}^d)^{s-1}\to \mathbb{C}$, $\psi_j: \mathbb{C}^d\to \mathbb{C}$ in the case of $m\le s+1$. All non-elementary solutions are described by the Weierstrass sigma-function. Previously, such results were known only for $s=2$, $m=1,2$, as well as for $d=1$, $s=2,3$. 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00638_а


DOI: https://doi.org/10.33048/semi.2019.16.030

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Bibliographic databases:

Document Type: Article
UDC: 517.965, 517.583
MSC: 39B32, 33E05
Received January 30, 2019, published April 5, 2019

Citation: A. A. Illarionov, “Solution of functional equations related to elliptic functions. II”, Sib. Èlektron. Mat. Izv., 16 (2019), 481–492

Citation in format AMSBIB
\Bibitem{Ill19}
\by A.~A.~Illarionov
\paper Solution of functional equations related to elliptic functions.~II
\jour Sib. \`Elektron. Mat. Izv.
\yr 2019
\vol 16
\pages 481--492
\mathnet{http://mi.mathnet.ru/semr1072}
\crossref{https://doi.org/10.33048/semi.2019.16.030}


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