Entire solutions of a system of difference equations. (Solutions entières d’un système d’équations aux différences.) (French) Zbl 0796.39006

In answer to a question of D. W. Masser [Prog. Math. 31, 173-190 (1983; Zbl 0549.32002)] we prove that, for almost all systems of difference equations \(\sum_{0 \leq m \leq M} A_ m (z) f(z+\alpha m) = \sum_{0 \leq n \leq N} B_ n(z) f(z+ \beta n)=0\), where \(A_ m\) and \(B_ n\) are polynomials and \(\alpha, \beta \in \mathbb{C}^*\) are \(\mathbb{R}\)- linearly independant, any solution \(f\) which is an entire function is the quotient of an exponential polynomial by a polynomial. We give a similar result when the second relation is replaced by a differential equation \(\sum_{0 \leq n \leq N} B_ n (z)f^{(n)} (z)=0\).
Reviewer: J.-P.Bézivin


39A10 Additive difference equations
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B32 Functional equations for complex functions


Zbl 0549.32002
Full Text: DOI Numdam EuDML


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