Thoughts on hyper-operations of rational but non-integer orders?

[VSO]

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?

[Gottfried]

I am not sure I get your problem correctly.           

Take the function \( f: b^z \) as to be iterated, with, say \( b=sqrt(2) \) .             
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 \( z_0 \) .              
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 \( z_1=b^{z_0} \). Then repeat it with a third plane, again 10 cm above, marking \( z_2=b ^{b^{z_0}} \) .
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 \( z_0 \) 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 \( z_0 \) 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

[tommy1729]

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

[Catullus]

Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?

Although, each level of the Ackermann function is primitive resursive. The entire Ackermann function itself can not be de-recursed.

[MphLee]

Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?

Although, each level of the Ackermann function is primitive resursive. The entire Ackermann function itself can not be de-recursed.

This comment doesn't address VSO's question. Ackermann function do not coincides with hyperoperations. Secondly, hyperoperations need not to be defined recursively. Third point: ackermann function, goodstein function (natural hyperoperations), if considered as functions over the natural numbers are recursive. Are defined recursively.

It is not fully clear what do you mean by de-recursed. Anyways, if the question is if it's possible to give an analytical, non recursive, expression to the ackermann function/hyperoperations, the answer is yes.
JmsNxn gave one expression for that. The real question is if that analytical representation has desired properties: does it satifies a functional equation? Is it smooth? Holomorphic?