So I've been running through my head a question. But in order to resolve that question, I need to better understand Kneser's construction. The premise of this post is to talk about constructing pentation from Kneser's construction.

Now, I know that,

\(

\text{tet}_{\text{Kneser}}(s) : \mathcal{H} \to \mathbb{C}\\

\)

Where \( \mathcal{H} = \{\Im(s) > 0\} \). And I know it tends to a fixed point \( L \) as \( \Re(s) \to -\infty \). (Or is it multiple fixed points?) Either way, this convergence must be geometric. Giving us, what I'll call,

\(

\Phi_{\text{Kneser}}(s) = \Omega_{j=1}^\infty \text{tet}_{\text{Kneser}}(s-j+z) \bullet z : \mathcal{H} \to \mathbb{C}\\

\)

Satisfying the functional equation,

\(

\Phi(s+1) = \text{tet}(s+\Phi(s))\\

\)

And it tends to \( L \) as \( s \to -\infty \) exactly like \( \text{tet}(s-1 + L) \).

Proving this converges will be a little troublesome if we don't have good control over Kneser's solution; but for the moment assume it's good enough. It'll definitely work for large enough \( T \) with \( \Re(s) < -T \). The idea then being,

\(

\text{slog}^{\circ n} \Phi(s+n) \to \text{pent}(s)\\

\)

Now, I know what you're thinking. Won't the same thing happen that happened with tetration happen with this method? And I'd say probably. Now instead of looking for \( \log(0) \)'s though, we'd be looking for \( \text{slog}(L) \)'s. So where ever \( \Phi(s_0) = L \) we get ourselves into a whole lot of trouble. The only real benefit I can think in this circumstance is that \( \Phi \) is not periodic. And that \( \Phi(s) \to L \) but it shouldn't equal \( L \), or at least, it should be well controlled where it does. This depends on Kneser though, does it ever attain the value \( L \) other than at \( -\infty \)?

I really wish there was more supplemental literature on Kneser's construction other than what's available on this forum... -_-

Nonetheless, I think this might have a better chance at converging than my tetration function constructed with \( \phi \). It's just a hunch though.

Now, I know that,

\(

\text{tet}_{\text{Kneser}}(s) : \mathcal{H} \to \mathbb{C}\\

\)

Where \( \mathcal{H} = \{\Im(s) > 0\} \). And I know it tends to a fixed point \( L \) as \( \Re(s) \to -\infty \). (Or is it multiple fixed points?) Either way, this convergence must be geometric. Giving us, what I'll call,

\(

\Phi_{\text{Kneser}}(s) = \Omega_{j=1}^\infty \text{tet}_{\text{Kneser}}(s-j+z) \bullet z : \mathcal{H} \to \mathbb{C}\\

\)

Satisfying the functional equation,

\(

\Phi(s+1) = \text{tet}(s+\Phi(s))\\

\)

And it tends to \( L \) as \( s \to -\infty \) exactly like \( \text{tet}(s-1 + L) \).

Proving this converges will be a little troublesome if we don't have good control over Kneser's solution; but for the moment assume it's good enough. It'll definitely work for large enough \( T \) with \( \Re(s) < -T \). The idea then being,

\(

\text{slog}^{\circ n} \Phi(s+n) \to \text{pent}(s)\\

\)

Now, I know what you're thinking. Won't the same thing happen that happened with tetration happen with this method? And I'd say probably. Now instead of looking for \( \log(0) \)'s though, we'd be looking for \( \text{slog}(L) \)'s. So where ever \( \Phi(s_0) = L \) we get ourselves into a whole lot of trouble. The only real benefit I can think in this circumstance is that \( \Phi \) is not periodic. And that \( \Phi(s) \to L \) but it shouldn't equal \( L \), or at least, it should be well controlled where it does. This depends on Kneser though, does it ever attain the value \( L \) other than at \( -\infty \)?

I really wish there was more supplemental literature on Kneser's construction other than what's available on this forum... -_-

Nonetheless, I think this might have a better chance at converging than my tetration function constructed with \( \phi \). It's just a hunch though.