TITLE: Concrete example of an entire function wanted
QUESTION [6 upvotes]: Let $X$ be the space of entire functions on $\mathbb{C}$ endowed with the topology of uniform convergence on compact sets. Let $a$ be a nonzero complex number. Let $T: X\to X$ be defined by $T(f)(z)=f(z+a)$. By a result of Birkhoff (see Dynamics of Linear Operators, by F. Bayart and E. Matheron, p. 3), there exists an $f\in X$ such that $O(f,T)=\{T^nf: n=0,1,2,\cdots\}$ is dense in $X$. 
Does anyone know a concrete example of such $f$?

REPLY [2 votes]: Except the Riemann Zeta function and similar functions (which are not entire functions), no concrete example is known so far.
For a survey of known results, I suggest you take a look at the recent papers by Paul Gauthier, e.g. this one. After Theorem 13, there is a remark saying that no concrete example of universal entire functions is known.