Solution of functional equations related to elliptic functions
type: Item
instance of: scholarly article
title: Solution of functional equations related to elliptic functions
zbMATH Open document ID: 1397.39015
DOI: 10.1134/S0081543817080065
author: A. A. Illarionov
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2018-08-16T00:00:00Z
review text: The author solves the functional equations \[ f(x+y)g(x-y)=\sum_{j=1,2,3}\alpha_j(x)\beta_j(y) \] and \[ f_1(x+z)f_2(y+z)f_3(x+y-z)=\sum_{j=1,2,3,4}\phi_j(x,y)\psi_j(z). \] All the functions are unknown and entire. These equations are connected with addition theorems for elliptic functions, as for instance, the relation \[ \sigma(x + y)\sigma(x- y) = \sigma^2(x)\sigma^2(y)(\wp(y) -\wp(x)). \] The author proves that the solutions of the functional equations are elliptic functions and quasi-polynomials \(\sum_{j=1}^s P_j(z) \exp (\lambda_j z)\), where \( P_j(z)\) are polynomials.
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 6919526
zbMATH Keywords: functional equation
zbMATH Keywords: entire solutions
zbMATH Keywords: addition theorem for elliptic functions
reviewed by: Vladimir Mityushev
MaRDI profile type: Publication
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full work available at URL: s0081543817080065
OpenAlex ID: W2789278750
description: scientific article; zbMATH DE number 6919526