User blog:Fejfo/Generalised Factorial

I'm sure I'm not the first one to think about generalising the factorial function. There may even be a wikipage about it but I want to share my own thoughs about it.

Regular factorial
To generalise something you first need to choose the fundamental property you want to generalise. To me the factorial function, fundamentally takes a binairy function

$$\times$$ and turns in into a unairy function using a the recursive relation

$$(n+1)! = (n+1) \times n!$$ With the bases case

$$1! = 1$$

Generalising
So we have a way to turn a binairy function into a factorial like unairy function. But to generalise we need a way to iterate the prosses, ie a way to turn the unairy function back into a binairy one. To me the most natural way is to use a simular recursive relationship:

$$n!(m+1) = (n!m)!$$ With the base case

$$n!0 = n$$

So we obtain:

$$n!m = n!...!$$ with m factorial signs

Definition
All that's left to do is to choose a base function to rigurously define the function. To me choosing the (unairy) succesor function feels natural.

$$n(0) = n+1$$ $$n(k)(m+1) = (n(k)m)(k)$$ $$n(k)0 = n $$ $$(n+1)(k+1) = (n+1) (k) (n(k+1)) $$ $$1 (k) = 1$$ k>0 It isn't to hard to see the growth rate is caped at $$f_\omega$$, you couldn't really expect anything stronger. $$n(0)m = n+m$$ $$n(1) = (n+1)n/2 = $$ the n-th triangle number

After this exact values become hard to express. But doing the same process starting from different funtions can still yield intresting results:
 * succesor -> addition -> triangle numbers
 * multiplication -> factorial -> iterated factorial
 * exponentiation -> $$n^...^4^3^2^1$$

It seems to be less interesting than I had hoped.

I just realised: you could see this hole process as making a ternairy function from a unairy one/binairy one. You could then turn the ternairy function into a binairy one in a factorial like one. continuing to turn functions into n-airy funtions we could actually reach decently sized numbers.