On to C^\infty--and attempts at C^\infty hyper-operations

[tommy1729]

So I posted a proof of \( C^\infty \) before I started working today...

I am in the process of rewriting my entire paper to focus on \( C^\infty \) hyper-operations; whereby it's a long proof by induction. But the initial step is to prove that this tetration is \( C^\infty \). Now I can most definitely show this tetration is \( C^\infty \); the trouble I'm having is making the proof as general as possible; so that we can create a proof by induction showing \( e \uparrow^k t \) is \( C^\infty \).

Hi James,  
I like your paper.  I would suggest generating an infinite sequence of entire \( \phi_n \) functions, perhaps defined as follows; this is slightly modified from your approach where this \( \phi_2(s) \) = JmsNxn phi(s+1)
we could start with 
\( \phi_1(s)=\exp(s) \)
\( \phi_2(s)=\exp(\phi_2(s-1)+s);\; \) this \( \phi_2(s) \) asymptotically approaches exp(s) as \( \Re(s) \) gets arbitrarily negative, 
\( \phi_n(s)=\phi_{n-1}(\phi_{n}(s-1)+s);\; \) \( \phi_n(s) \) also asymptotically approaches exp(s) as \( \Re(s) \) gets arbitrarily negative

James has proven that \( \phi_2(s) \) is entire, and I think each of these phi functions is also entire, and each \( \phi_n(s) \) would probably lead to an \( e\uparrow^n(s)\; \) function which is also \( C^\infty \) only defined at the real axis; details tbd...

\( \phi_3(s)=\phi_{2}(\phi_{3}(s-1)+s) \) probably grows more like pentation.
Notice the similarity to the superfunctions of the previous function in the list.

So that would probably fail to get another c^oo solution to tetration but rather a c^oo solution to pentation or higher.
Unfortunately probably not analytic either.

Generalizing to fractional index n is then probably similar to the classic ' semi-super ' function type questions.

So sorry but .. I am not convinced of its usefullness.

***

\( f_n(s)=f_{n-1}((f_{n}(s-1)+s)/2) \) **could** however converge to f(s) = s + 1 ( the successor function !) for appropriately defined f_1(s).

Maybe that could be usefull for some kind of hyperoperator ?
However going to negative index n does not seem to give interesting results ( only linear functions ?).

Ofcourse many variants of the above can be considered and the question is very vague and open.
But it is not certain in what direction we should proceed .. or is it ??

It does not seem simpler than generalizing ackermann , making ackermann analytic etc 

Regards

tommy1729