During a talk about his article [Gra1] in Oberwolfach, D.W. Masser asked F. Gramain the following question : are the entire functions solutions of the difference equations system
In a first article [BéGra1], J.-P. Bézivin and F. Gramain gave a negative answer to this question considering the example of the function , and showed that if belongs to and if the coefficients of one of the two equations are constant, then is an exponential polynomial. Moreover, they proved, still assuming that belongs to , 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 an entire function entière solution of the system (1) then 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 and for every exponential polynomial , there exists a finite relation of linear dépendence over between the shifts of by (we say that is a recurrent step for the entire function -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 by by derivatives of , an entire solution is still the quotient of an exponential polynomial by a polynomial. The algorithm slightly modified allows again to determine the entire solutions of the system.