Question 30
Let $\mathbb{Z}_{\gt 0}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ such that
$$
f^{b f(a)}(a+1)=(a+1) f(b)
$$
holds for all $a, b \in \mathbb{Z}_{\gt 0}$, where $f^{k}(n)=f(f(\cdots f(n) \cdots))$ denotes the composition of $f$ with itself $k$ times.
$$
f^{b f(a)}(a+1)=(a+1) f(b)
$$
holds for all $a, b \in \mathbb{Z}_{\gt 0}$, where $f^{k}(n)=f(f(\cdots f(n) \cdots))$ denotes the composition of $f$ with itself $k$ times.