Introduction

During a talk about his article [Gra1] in Oberwolfach, D.W. Masser asked F. Gramain the following question : are the entire functions $${f}$$ solutions of the difference equations system  

$$\left\{ \begin{array} {c} \sum_{0 \leq m \leq M} P_m(z)f(z+... ...sum_{0 \leq n \leq N} Q_n(z)f(z+n\beta )= 0,\end{array} \right.$$ (1)

(where $(P_m)_{0 \leq m \leq M}$ and $(Q_n)_{0 \leq n \leq N}$ are finite sequences of $\mathbb C[z]$ with $P_M Q_N \neq 0$, $\alpha$ and $\beta$ being complex numbers linearly independent over $\mathbb R$) necessarily exponential polynomials, i.e. functions of the shape $\sum_{i=1}^{K} a_i (z) e^{\alpha_i z}$ with $${ a_i}$$ in $\mathbb C[z]$ and $\alpha_i$ in $\mathbb C$ for all $${i}$$ ?

In a first article [BéGra1], J.-P. Bézivin and F. Gramain gave a negative answer to this question considering the example of the function $${f(z)=(e^z-1)/z}$$, and showed that if $\alpha/\beta$ belongs to $\mathbb C\backslash \mathbb R$ and if the coefficients of one of the two equations are constant, then $${f}$$ is an exponential polynomial. Moreover, they proved, still assuming that $\alpha/\beta$ belongs to $\mathbb C\backslash \mathbb R$, a serie of partial results that allowed us to conjecture that every entire solution of a system with polynomial coefficients is the quotient of an exponential polynomial by a polynomial. This conjecture has been proved with other results in a second article [BéGra2] :

Théorème  [J.-P. Bézivin- F. Gramain]
Let $${f}$$ an entire function entière solution of the system (1) then $${f}$$ is the quotient of an exponential polynomial by a polynomial.

In fact, we can replace the right members of (1) by exponential polynomials. We only have to notice, as in the "Lemme 2.5" of [BéGra1], that for every complex number $\alpha$ and for every exponential polynomial $\Phi$, there exists a finite relation of linear dépendence over $\mathbb C$ between the shifts of $\Phi$ by $\alpha$ (we say that $\alpha$ is a recurrent step for the entire function $\Phi$ -it is a definition due to D. Masser).

In the last part of [BriHab], we present an algorithm that allows to determine the entire solutions of the systèm (1).

Moreover, J.-P. Bézivin and F. Gramain proved in [BéGra2] that if we replace, in the second equation of (1), the shifts of $${f}$$ by $\beta$ by derivatives of $${f}$$, an entire solution $${f}$$ is still the quotient of an exponential polynomial by a polynomial. The algorithm slightly modified allows again to determine the entire solutions $${f}$$ of the system.  

$$\left\{ \begin{array} {c} \sum_{0 \leq m \leq M} P_m(z)f(z+... ... \sum_{0 \leq n \leq N} Q_n(z)f^{(n)}(z)= 0,\end{array} \right.$$ (2)

where $(P_m)_{0 \leq m \leq M}$ and $(Q_n)_{0 \leq n \leq N}$ are finite sequences of $\mathbb C[z]$ with $P_M Q_N \neq 0$, $\alpha$ being a non zero complex number.