Quote:what lies between tetration and pentation ?Do we know what lies between addition and multiplication, or multiplication and exponentiation? I would be happy to know those first. I assume they would be simpler to find, but I can also imagine that they would be equally difficult to find.

Quote:Question : if we say S^[a+b](f(x)) = S^[a](S^[b](f(x)) = S^[b](S^[a](f(x))I found an answer to part of your question. By that I mean I was able to find S^[1/2](exp(x)):

Then what is S^[1/2](f(x)) ? Or what is S^[1/2](exp(x)) ?

By definition:

So we are trying to find some function such that

If we define such that

then

, where W is the Lambert W function

There is your half-superfunction of exp(x).