Sur les équations aux différences en une variable. (Difference equations in one variable) (Q1586377)

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.