Generalized arithmetic operator

[tommy1729]

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.

Question : if we say S^[a+b](f(x)) = S^[a](S^[b](f(x)) = S^[b](S^[a](f(x))
Then what is S^[1/2](f(x)) ? Or what is S^[1/2](exp(x)) ?

I found an answer to part of your question. By that I mean I was able to find S^[1/2](exp(x)):

By definition:
\( S^{1/2}(S^{1/2}(e^x))=e^x \)

So we are trying to find some function \( f \) such that
\( f(f(x))=e^x \)

If we define \( b \) such that
\( f(x)=e^{bx} \)

then
\( f(f(x))=e^{be^{bx}}=e^x \)

\( \Rightarrow be^{bx}=x \)

\( \Rightarrow bx\cdot e^{bx}=x^2 \)

\( \Rightarrow bx=W(x^2) \), where W is the Lambert W function

\( \Rightarrow b=W(x^2)/x \)

\( \Rightarrow f(x)\equiv S^{1/2}(e^x)=e^{bx}=e^{W(x^2)} \)

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

Sorry for not using tex before but

By definition:
\( S^{1/2}(S^{1/2}(e^x))=S^{1/2+1/2}(e^x)=S(e^x)=sexp(x) \)

that is sufficient to see your answer is wrong ...
Sorry.

regards

tommy1729