Notation of recursive function with multiple arguments

Question

I want to formally describe an algorithm $f : E \times P \to P$ which you can use as an argument for itself. Right now, I have the following definition:

\begin{equation} \begin{split} P_{e_{n+1}} &= f(e_n, P_{e_n}) \\ &= f(e_n, f(e_{n-1}, f(e_{n-2}, \dots f(e_1, P_{e_1})))) \end{split} \end{equation}

The formula should show that you only need one initial $P_{e_1} \in P$ and a set of elements $e_{x} \in E$ to generate $P_{e_{n+1}}$. Is there a name for such a function or a notation that is able to indicate such a characteristic?

Answer 1

Hint: You might want to have a look at Ackermann functions.