Hypertranscendence and linear difference equations, the exponential case

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