Grzegorczyk hierarchy vs Iterated differential equations?

[MphLee]

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 and the hierarchy :

1-Any function in is computable in

2-If then is the extension to the reals of some then

3-the converse holds: if is a function on the naturals of rank it has an extension in

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The interesting thing is that the various levels of 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- is defined as follow

I-the constants ,, and , the projection functions, are in

II- is closed composition and linear integration

in a recursive way we define

III- contains the functions in

IV- 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

V- is closed under composition and linear integration

and

if
if