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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!^5[[0]]

Examples

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

Nested Hyperfactorial Array Notation

Nested Factorial Notation + Hyperfactorial Array Notation

m![@ 1,n+1 @] = m![@ [@ 1,1 @],n @]

Up Chained Arrow Notation

Up Arrow Notation + Chained Arrow Notation

5. \(a \circ\uparrow (b+1) = a \circ (a \circ\uparrow b)\)

6. \(a \circ\rightarrow b \circ\rightarrow c = a \circ\uparrow^c b\)

Torsius' Factorial Array Notation

Torsius' Factorial + Hyperfactorial Array Notation

n!1 = n!

n!m = n!(m-1)*(n-1)!(m-1)...2!(m-1)*1!(m-1)

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