User blog:Ynought/Function iterator?

So i was thinking and i came up with this:

\(\begin{eqnarray*} F:\mathbb{N}\mapsto\mathbb{N} \end{eqnarray*}\) and \(F\) can also be a function which is a function that has another operatuion on it(example later)(but you have too put that function in a pair of these brackets \(\{ \}\) ) and the resultion function will be highlighted by a pair of these \([ ]) brackets

\(F+0(n)=[F](n)\)

\(F+k(n)=[F^n k-1](n)\)

\(F+F(n)=[F+F[n]](n)\)

so far \(F+n(n)\) reaches \(f_\omega (n)\) where \(f_0(n)=F(n)\)

and i will put : \(F[n]=F(n)\)

so then i will continue:

\(F\times 0(n)=[F](n)\)

\(F\times k(n)=[\{F\times k-1\}+n](n)\)

so far \(F\times n(n)\) reaches \(f_{\omega^2} (n)\) where \(f_0(n)=F(n)\)

then:

\(F\times F(n)=[F\times F[n]](n)\)

\(F^0(n)=[F](n)\)

\(F^k(n)=[\{F^{k-1}\}\times F] (n)\)

\(F^F(n)=[F^{F[n]}](n)\)

<p data-parsoid="{"dsr":[8455,8549,0,0]}">Then \(F^{F^{...^{F}}}(n)\approx f_{\varepsilon_0}(n)\) where \(f_0(n)=F(n)\)

<p data-parsoid="{"dsr":[8551,8657,0,0]}">And i would like to push it up to \(f_{\varphi(\omega,0)}(n)\) so i came up with "Function arrow notation"

<p data-parsoid="{"dsr":[8659,8662,0,0]}">So:

<p data-parsoid="{"dsr":[8664,8691,0,0]}">\(F\uparrow k(n)=[F^k](n)\)

<p data-parsoid="{"dsr":[8693,8721,0,0]}">\(F \uparrow^i 1(n)=[F](n)\)

<p data-parsoid="{"dsr":[8723,8761,0,0]}">\(F \uparrow^1 k(n)=[F\uparrow k](n)\)

<p data-parsoid="{"dsr":[8763,8824,0,0]}">\(F \uparrow^i k(n)=[F\uparrow^{i-1}\{F\uparrow^i k-1\}](n)\)

<p data-parsoid="{"dsr":[8826,8911,0,0]}">And if you come across an \(F\) that you have to reduce to \(F-1\) then do \(F[n]-1\)

<p data-parsoid="{"dsr":[8913,9066,0,0]}">And if i am right then it is basically "Function catched point function" so that \(F\uparrow^F F(n)\approx f_{\varphi(\omega,0)}(n)\).That is it for now.