Definition:Superfunction

Superfunction comes from the iterations of some given function, called transfer function; the argument indicates, how many times the transfer function should be iterated; and the result of such iterations is value of the superfunction.

Definition
Let $C\subseteq\mathbb C$ be some domain in the complex plane such that $z\!+\!1\in C \forall z \in C$

Let $D\subseteq\mathbb C$ be some domain in the complex plane.

Let $F: D\mapsto D$ be holomorphic function.

Let $H: C\mapsto D$ be holomorphic function.

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

Then $F$ is called superfunction of function $H$, and function $H$ is called transfer function of $F$.

Examples
for any $b\in \mathbb C$, the function $z \mapsto z \times b$ is superfunction of $z\mapsto z+b$, and $z\mapsto z+b$ is transfer function of $z \mapsto z \times b$.

Function $\exp$ is superfunction of $z \mapsto e z$.