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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

MSC:

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

Citations:

Zbl 0549.32002
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References:

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