Generalization of Hyperoperations / fractional operations

[benjayk]

Hi everybody!
I recently came across the hyperoperation sequence which extends the sequence of operations x+y, x*y, x^y to operations x[n]y, which are recursively defined as "the previous operation applied y times on x".
So I asked myself: Can this be generalized to positive rational (or even negative /irrational/complex) numbers for n (yes, for n!). This may seem weird, but why not? After all new structures in math are often discovered by asking "weird" questions, like what is the root of -1 (complex numbers) or is there an extension of the factorial to real numbers (gamma function), etc...

Has someone tried to define / calculate / study such "fractional" operators? I haven't found anything substantial on the internet, but maybe I don't know the right term to search for?
Is it even possible to find an extension that makes sense (that is, it should satisfy x[n]x=x[n+1]2 and the function x[n]y should probably be a monotonic function for all combinations of positive integers x and y and maybe even infinitely often differentiable)? If not, why not?
There is certainly no easy way to express these operations with existing operations / functions, right?

It would surprise me if this hasn't been researched yet, as the operators are essential in mathematics and we try to generalize existing structures in math to understand new relations. I would guess that fractional operators (if they exists) may yield understanding of existing relationships (maybe some integrals that can not be definied in terms of existing functions could be defined by those operators or things like that) and possibly may be used to express new relationships (maybe even physical ones).

If this is not being studied and an open question, do you think it somehow unimportant or unintersting or why does almost no one try to define fractional operators?

 

[g_edgar]

http://en.wikipedia.org/wiki/Tetration

 

[benjayk]

I don't get what this has to do with my question. Tetration is the hyperoperation with n=4, which is a positve integer. I asked for hyperoperations (or the generalization thereof) where n is not a positive integer.

I have to make a correction, I wrote: "x[n]y should probably be a monotonic function for all combinations of positive integers x and y", but this is not true even for all positive whole numbers n (eg 1+2=3, yet 1*2=2). So it should read "x[n]y should probably be a monotonic function for all combinations of positive integers x and y bigger or equal to 2".

 

[GFR]

Hi, benjayk,
I am pleased to see your interest in the hyperoperation hierarchy and, in particular in its extension to the non-integer ranks. Some half-integer ranks have been investigated, such as:
y = b [k] x, with k = 0.5 (halfation), and k = 1.5 (sesquition or sesquation).
In particular, sesquition is a half-way hyperoperation between addition and multiplication and seems to be justified by the existence of the arithmetic-geometric mean, called also the Gauss Mean, obtainable through the "complete helliptic integral of the first kind".
Plese see: Hyperoperations for Science and Technology, (K. Rubtsov, G. Romerio), Lambert Academic Publishing, 2010 (ISBN 978-3-8443-1516-5), pages 98-104.
The existence of a midway operation between addition and multiplication had been imagined by Prof E. Williams, London Scool of Economics. The point is that, if the arthmetic-geometric mean exists, the sesquition should also ... perhaps exist. Therefore, an overall half-integer hyperoperation rank may also be possible. However, a lot of work needs still to be done.
Welcome in this extraorrdinary entreprise!
GFR

 

[Dr. Seafood]

I think the fractional differentiation operator would be interesting to you: it's a function D^1/2 with the property that, for a differentiable real function of x, D^1/2(D^1/2 f(x)) = f'(x), the derivative of the function f(x).

http://en.wikipedia.org/wiki/Fractional_calculus

 

[GFR]

Thank you! I am going to look into it. GFR