Difference equations associated with groups, representative functions.
type: Item
instance of: scholarly article
title: Difference equations associated with groups, representative functions.
zbMATH Open document ID: 1062.39001
author: Nicolas Marteau
publication date: 2004-12-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116115
review text: Let \(G\) be a group and \(H\) be its subgroup. The function \(f: G\to\mathbb{C}\) is called right \(H\)-representative (respectively left) if the complex vector space generated by the right translates \(f_h\) (respectively left), \(h\in H\) is finite-dimensional. If \(H = G\), \(f\) is called representative. The author, inspired by the results of \textit{J.-P. Bézivin} and \textit{F. Gramain} [Ann. Inst. Fourier 43, No. 3, 791--814 (1993; Zbl 0796.39006)], establishes the link with the system of difference equations with constant coefficients. Let \(G\) be a connected complex Lie group and \(H\) be its subgroup. If every complex valued entire function on \(G\) and right or (left) \(H\)-representative is representative, then every entire \(H\)-invariant function is constant. The converse is also true if \(G\) is abelian or nilpotent.
Mathematics Subject Classification ID: 39A05
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 2123571
zbMATH Keywords: difference equations
zbMATH Keywords: Lie groups
zbMATH Keywords: system of difference equations
zbMATH Keywords: constant coefficients
reviewed by: Bolis Basit
MaRDI profile type: Publication
OpenAlex ID: W2322737531
cites work: Q4833694
cites work: Entire solutions of a system of difference equations
cites work: Solutions entières d'un système d'équations aux différences. II. (Entire functions solutions of a system of difference equations. II.)
cites work: Sur les fonctions entières à double pas récurrent. (Entire functions with double recurrent step.)
cites work: Periodic meromorphic functions
cites work: Q3844558
cites work: Q3856819
cites work: Q3342868
cites work: Q5669004
cites work: Linear algebraic groups
cites work: Q4527707
cites work: Complex analytic geometry of complex parallelizable manifolds
DOI: 10.5802/AIF.2022
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2123571