Hypertranscendence and linear difference equations, the exponential case

In this paper we study meromorphic functions solutions of linear shift difference equations in coefficients in $\mathbb{C}(x)$ involving the operator $\rho :y(x)\mapsto y(x+h)$, for some $h\in {\mathbb{C}}^{\ast }$. We prove that if $f$ is solution of an algebraic differential equation, then $f$ belongs to a ring that is made with periodic functions and exponentials. Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer.