On the space of entire solutions of a system of difference equations

In this Note, we prove that the space of entire solutions of a system F(z + 1) = A(z)F(z), F(z + alpha) = B(z)F(z), where a epsilon R and A and B are (n, n)-matrices whose coefficients ai e f unctions of z, is of finite dimension over C. We give the following applica tion: entire functions f : C --> C which are solutions of the system Sigma (j=J)(j=0) a(j)f(z + j) = Sigma (k=K)(k=0) b(k)(z)f(z + k alpha)=0, where alpha epsilon B, a(0)a(J) not equal 0, the b(k) are non-zero, meromor phic and tau -periodic functions (tau epsilon Q), are exponential polynomia ls. (C) 2000 Academie des sciences/Editions scientifiques et medicales Else vier SAS.