# Tetration Forum

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Inspired by JmsNxn's thread (http://math.eretrandre.org/tetrationforu...39#pid7139) about the continuum sum I repost this obsevation about the link between the fractional calculus and the Hyperoperations.
I guess that there can be interesting links... and probably is not the wrong way to approach the problem. I just found some results about something similar.

M. Campagnolo, C. Moore -Upper and Lower Bounds on
Continuous-Time Computation

In this text I found a relation betwen a hierarchy of real valued function and the Grzegorczyk hierarchy.

The interesting relations are betwen a hierarchy called $\mathcal{G}_n+\theta_k$ and the hierarchy $\mathcal{E}_n$:

Quote:1-Any function in $\mathcal{G}_n+\theta_k$ is computable in $\mathcal{E}_n$

2-If $f\in \mathcal{G}_n+\theta_k$ then $f$ is the extension to the reals of some $f^{*}:\mathbb{N}\rightarrow\mathbb{N}$ then $f^{*}\in \mathcal{E}_n$

3-the converse holds: if $f$ is a function on the naturals of rank $n$ it has an extension in $\mathcal{G}_n+\theta_k$

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The interesting thing is that the various levels of $\mathcal{G}_n+\theta_k$ are defined via iterated solution of a special kind of functional equation...and that maybe can be linked with your knowledge in this field...

Definition-$\mathcal{G}_3+\theta_k$ is defined as follow

Quote:I-the constants $0$,$1$,$-1$ and $\pi$, the projection functions, $\theta_k$ are in $\mathcal{G}_3+\theta_k$

II-$\mathcal{G}_3+\theta_k$ is closed composition and linear integration
in a recursive way we define $\mathcal{G}_{n+1}+\theta_k$
Quote:III- $\mathcal{G}_{n+1}+\theta_k$ contains the functions in $\mathcal{G}_{n}+\theta_k$

IV- $\mathcal{G}_{n+1}+\theta_k$ in we can find all the solutions to the equation (2) in this text ( http://languagelog.ldc.upenn.edu/myl/DK/...oMoore.pdf ) applied to the functions in $\mathcal{G}_{n}+\theta_k$

V-$\mathcal{G}_{n+1}+\theta_k$ is closed under composition and linear integration

$\theta_k(x):=x^k\theta(x)$ and

$\theta(x):=0$ if $x \le 0$
$\theta(x):=1$ if $x \gt 1$