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Thoughts on hyper-operations of rational but non-integer orders?
#1
I can't seem to find the right angle to approach this concept intuitively. Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?
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#2
I am not sure I get your problem correctly.           

Take the function as to be iterated, with, say .             
Assume one plane on a math-paper and look for easiness only the lines and their crossings of the coordinate-system of integer complex numbers .              
Now take another paper, position it 10 cm above and for every point of the crossings (and ideally also of the lines) mark the values of . Then repeat it with a third plane, again 10 cm above, marking .
After that, try to connect the related points of the zero'th, the first and the second plane by a weak string, say a spaghetti or so. Surely except of the fixpoints in it shall be difficult to make a meaningful and smooth curve - and in principle it seems arbitrary, except at the fixpoints, where we simple stitch a straight stick through the iterates of the at the fixpoint.           
Of course the spaghetti on the second level is then no more arbitrary but must be - point for point - be computed by one iteration. But the spaghatti in the first level follow that vertically orientated curve, where a fictive/imaginative plane of paper is at fractional heights and the fractional iterates would be the marks on the coordinate-papers at the "fractional (iteration) height".                                                     

I'd liked to construct some physical example, showing alternative paths upwards between the fixed basic planes, with matrial curves made by an 3-D-printer, but I've not yet started to initialize the required data.

But I think, that mind-model alone makes it possibly already sufficiently intuitive for you.


A somewhat better illustration is in my answer at MSE, see https://math.stackexchange.com/a/451755/1714
Gottfried Helms, Kassel
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#3
I think the OP refers to concepts like , what i called " semi- super " operators.

Like the semisuper operator of the semisuper operator of f(x) is the super of f(x).

This is extremely difficult.

Do not confuse with the functional half-iterate of the superfunction.

Regards 

Tommy1729
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