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

Hypertranscendence and linear difference equations, the exponential case