generalizing the problem of fractional analytic Ackermann functions

[tommy1729]

if perhaps I read you wrong, and maybe you meant to say

\( (f^{\diamond -n} \diamond f^{\diamond n})(x)= f(x) \)

\( (f^{\diamond -n} \diamond f^{\diamond n+1}(g^{\diamond n + 1}(x) + 1))(x) = f^{\diamond 1}(g^{\diamond n + 1}(x) + 1) \)

this would imply inconsistency.

Which to me is no surprise, it was just suggested in a makeshift attempt to come up with a recurrence relation for fractional values of superfunction indexes. It was by no means canon.

It hardly makes the rest of my post "inconsistent" as if you read carefully I even asked [in reference to the diamond operator]:

So I'm wondering, is this just absurd? It's really the best I can come up with...

That being said, the concept of

\( g^{\diamond n}(f^{\diamond n}(x)) = x \)

\( f^{\diamond n}(x) = f^{\diamond n+1}(g^{\diamond n+1}(x) + 1) \)

is still perfectly consistent which was the main point of the post. And it is hardly wrong at first glance.

therefore I still can't help but think by your post that you mistook \( \diamond \) for \( \circ \)

??

first you say you would not be surprised if it was inconsistant.

then you continue , the rest of the post remains correct.

but everything is based on the beginning equations ?

as for composition , i know your diamond isnt composition.

but it does share many properties.

and you yourself gave a relation between the 0 th diamond of g and the compositional inverse of f.

i did not even mention composition.

assuming an equation is welldefined , it must be assumed that when inverses are introduced , they are introduced without switching branches.

hence if we do not loose information or do forbidden operations , we get a group structure.

this group structure has the diamond operator.

and you write - before your diamonds , suggesting it has inverses.

all diamond operators , f and g give a result that belongs to the same structure ( so we can continu taking diamonds ) hence all conditions are met for representation by groups.

if the diamond is your group operator and the inverse is also a group operator in that group ( what must be as explained above ) ,

then we can do left-inverses of the group operators or aka taking diamonds and anti-diamonds on the left side.

since by your first equation we can reduce the group operations (diamonds) we get at the contradion i posted.

( note what i did is more general then working with functional compositions of continu functions ... also a counterargument with integrals will fail because of the " + constant " after integrals )

so im far from convinced i made a mistake , if i did plz let me know clearly and keep in mind , you are the one proposing something so in fact it is actually up to you to prove that it exists.

i might apologize if im wrong.

but i doubt it that i am if you insist on all the equation you wrote down and x being real.

sorry for the emotional shit

regards

tommy1729