User blog:Googleaarex/Functions that combined

Dollar FGH
FGH + Dollar Function

Rule 3 of FGH: \(f_{[0]}(n)\) = \(f_n(n)\)

Examples
\(f_{[ [0]_2 ]}(3)\) = \(f_{[ [ [0] ] ]}(3)\) = \(f_{[ [ 3 ] ]}(3)\) = \(f_{\varepsilon_0}(3)\)

Nested Dollar Function Notation
Nested Factorial Notation + Dollar Function

5!0 = 5![5]

5![1[0]] = 5!^50

Examples
\(3![ [1] ]\) = \(3![ [0][0][0] ]\)