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

[sheldonison]

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...