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

[MphLee]

Since they are 2pi*I periodic with limiting behavior as real(z) grows arbitrarily negative of \( \exp(z) \), then each of these functions has a corresponding Schroeder function whose inverse \( \psi^{-1} \) is also entire with
\( \phi_n(z)=\psi_n^{-1}(\exp(z)) \)
\( \psi^{-1}_n(x)=x+\sum_{n=2}^{\infty}a_n\cdot x^n;\;\; \) there is a formal entire inverse Schroeder function for each phi_n function.

I'm probably missing some key piece of the puzzle (terminology). Are you talking about a kind of inverse Schroeder-like function right? A confortable abuse of name similar to how we can call \( \phi_{n+1} \) inverse Abel-like function of \( \phi_{n} \)?

In a strict sense, I don't see how \( \psi_{n} \) is a Schroeder function of \( \phi_{n} \) or of \( \phi_{n-1} \).

For this sequence I see this: define \( \sigma_{n}:=\psi_n^{-1} \) in your notation.

\( \sigma_{n+1}(z)=\phi_n(\ln z+\sigma_{n+1}(\frac{z}{e})) \)
\( \sigma_{n+1}(z)=\sigma_n(z\cdot e^{\sigma_{n+1}(\frac{z}{e})}) \)

That is "inverse Schroeder-like" in some sense.
\( \sigma_{n+1}(ez)=\phi_n(1+\ln z+\sigma_{n+1}(z)) \)
\( \sigma_{n+1}(ez)=\sigma_n(ez\cdot e^{\sigma_{n+1}(z)}) \)


ADD: it is possible to express phi_2 (an offset of JmsNxn's phi) as an infinite composition.

\( \phi_2(z)=\Omega_{j=0}^\infty \phi_1(z-j+w)\bullet w \) at \( w=0 \)

Is there a similar "closed form" for the other phi_n? Is it too optimistic to declare

\( \phi_{n+1}(z)=\Omega_{j=0}^\infty \phi_{n-1}(z-j+w)\bullet w \) at \( w=0 \)?