Definition:Superfunction

Definition
Let $C, D \subseteq \C$ with $z \in C \implies z + 1 \in C$.

Let $F: D \to D$ and $H: C \to D$ be holomorphic functions.

Let $H(F(z))=F(z+1)$ for all $z\in C$.

Then $F$ is said to be a superfunction of $H$, and $H$ is called a transfer function of $F$.

That is, superfunctions are iterations of transfer functions.