Sur les équations aux différences en une variable. (Difference equations in one variable)
type: Item
instance of: scholarly article
title: Sur les équations aux différences en une variable. (Difference equations in one variable)
zbMATH Open document ID: 0960.39002
author: Nicolas Marteau
publication date: 2000-11-13T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75465
review text: The author deals with the finite difference system of equations of the form \[ \sum_{j=0}^{J} a_j(z)f(z+\alpha_j)=0,\quad \sum_{k=0}^{K} b_k(z)f(z+\beta_k)=0, \tag{*} \] where \(J,K\in \mathbb N\), \(a_j,b_k\) are polynomials with complex valued coefficients and \(\alpha_k,\beta_k\in \mathbb C\). Under various additional assumptions on these polynomials and complex numbers, the properties of solutions to (*) are deduced. A typical result is the following statement. Theorem. Let \(\langle \alpha_j\rangle_{j=0}^J\cap \langle\beta_k\rangle_{k=0}^K=\{0\}\), where \(\langle \alpha_j\rangle\), \(\langle \beta_k\rangle\) denote the subgroups of \(({\mathbb C},+)\) generated by \(\alpha_j\) and \(\beta_k\), respectively. Further suppose that the sequences \(\Re(\alpha_j)\), \(\Im(\alpha_j)\), \(\Re(\beta_k)\) are strictly increasing and \(\Im(\beta_k)\) is strictly decreasing. Then every entire solution \(f\) of (*) is a ratio of an exponential polynomial and a polynomial.
reviewed by: Ondřej Došlý
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1528716
zbMATH Keywords: difference equations of one variable
zbMATH Keywords: elimination
zbMATH Keywords: finite difference system
zbMATH Keywords: entire solution
zbMATH Keywords: exponential polynomial
zbMATH Keywords: polynomial
MaRDI profile type: Publication
OpenAlex ID: W1999374470
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DOI: 10.5802/AIF.1801
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1528716