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Zbl 0960.39002
Marteau, Nicolas
Sur les équations aux différences en une variable. (Difference equations in one variable). (French)
[J] Ann. Inst. Fourier 50, No.5, 1589-1615 (2000). ISSN 0373-0956; ISSN 1777-5310/e

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 $J,K\in \Bbb N$, $a_j,b_k$ are polynomials with complex valued coefficients and $\alpha_k,\beta_k\in \Bbb C$. Under various additional assumptions on these polynomials and complex numbers, the properties of solutions to (*) are deduced. A typical result is the following statement. \par 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 $({\Bbb C},+)$ 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.
[Ondřej Došlý (Brno)]

MSC 2000:: *39A10 Difference equations

Keywords: difference equations of one variable; elimination; finite difference system; entire solution; exponential polynomial; polynomial

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