User blog:P進大好きbot/Kyodaisuutan System

I define a \(3\)-ary function \(\left(^{a,b}_{\ c} \right)\) with \(a,b,c \in \mathbb{N}_{> 0}\) by the following recursion:

\begin{eqnarray*} \forall a,b,c,d,e \in \mathbb{N}_{> 0}, \ \left(^{a,b}_{\ c} \right) = \lim_{x \to +0} \left( \frac{b+c}{2^{\frac{a-1}{x}}} + \frac{\frac{\left(^{d,b}_{\ b} \right)}{2^{\frac{c-1}{x}}} + \frac{\left(^{a,\left(^{a,b}_{\ 1} \right)}_{\ \ e} \right)}{2^{\frac{(c-e-1)^2}{x}}} + \frac{\left(^{a,b}_{\ c} \right)}{2^{\frac{c^{- \frac{1}{x}} + ((c-e-1)^2+1)^{- \frac{1}{x}}}{x}}}}{2^{\frac{(a-d-1)^2}{x}}} + \frac{\left(^{a,b}_{\ c} \right)}{2^{\frac{a^{- \frac{1}{x}} + ((a-d-1)^2+1)^{- \frac{1}{x}}}{x}}} \right) \end{eqnarray*}

This is a one-ruled large function which I submitted to a Japanese googology contest on large numbers smaller than or equal to \(f_{\omega}(10^{100})\), and the growth rate of \(\left(^{n,n}_{\ n} \right)\) is precisely \(f_{\omega}(n)\). I like this very much because the definition is very simple.

I call the technique to generate a one-ruled large function a Kyodaisuutan System. I note that Kyodaisuutan is a famous name of an artificial spirit of googology appearing in a Japanese ancient fairytale. That is why we call a large number a Kyodaisuu in Japanese.