Powerful arrow notation

Powerful arrow notation and its chained variant are an extension of the normal Up arrow notation.Mumuji, Powerful Arrow, Github pages, Retrieved at (UTC) 12:40 on August 30th in 2024. They were created by googology wiki user Mumuji.
==Definition==
Here is a copy of the description in the original source:
Note that all variables in this function are natural numbers. For PAN, the domain would be \(\mathbb{N}^3\). For the chained varaiant, we would have the domain as \(\mathbb{N}^n\) for \(n \in \mathbb{N}\)
\[\begin{eqnarray*} a \uparrow^n b = \left\{ \begin{array}{ll} {a}^b & (n = 1) \\ a & (n > 1, b = 1) \\ (a \uparrow^{n-1})^{b\uparrow^{n-1}b} (a \uparrow^n (b-1)) & (n > 1, b > 1) \end{array} \right. \end{eqnarray*}\]
The third rule is then clarified as follows:
\[f^{a+1}(x) = f^{a}(f^{a}(x))\],when \(a\ge0\) and the third rule simply has \(f(x) = a\uparrow^{n-1}(x)\).
The termination case is as follows: 
\[f^{0}(x) = f(x)\]
==Chained variant==
\(a\rightarrow^1 b = a^b\)
\(a\rightarrow^x b = f(a)\rightarrow^{x-1} f(a)\rightarrow^{x-1} f(a) \ldots \rightarrow^{x-1} f(a)\) where there are \(b\rightarrow^{x-1}b\) copies of \(f(a)\), where \(f(a) = a\uparrow^aa\) (refer above)
 \(\#\rightarrow^x 1 = \#\) (here, \(\#\) denotes an arbitrarily long part of the chain before the relevant terms)
\(\#\rightarrow^x 1\rightarrow^y a = \#\) \(\#\rightarrow^n a\rightarrow^1 b = \#\rightarrow^n (\#\rightarrow^n (a-1)\rightarrow^1 b)\rightarrow^1 (b-1)\)\(\#\rightarrow^n a\rightarrow^x b = \#\rightarrow^n a\rightarrow^{x-1} a\rightarrow^{x-1} a \ldots \rightarrow^{x-1} a\) where there are \(b\) copies of \(a\)
==Sources==
Category:Numbers and functions by Mumuji
Category:Notations
Category:Incomplete
Category:Arrow