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Is Fractionalation a thing? That is, if first order is addition, and second order is multiplication, is there a 1.5th order? a sqrt(2) order? a ith order operation?

Lets define a function A(o, x, y) where o is the order, and x and y are the arguments to the function.

A(1, x, y) would just be addition, x+y

A(2, x, y) would be multiplication, x*y, or x + x + x } y times

A(3, x, y) would be x^y or xxx } y times

A(4, x, y) would be tetration, i think it's denoted as ^y(x) or something? but x^x^x } y times

I raise the question, has this space been filled out? Can we take A(1.5, x, y)? or A(sqrt(2), x, y)? or A(2i, x, y)?

Is this something that has been studied? Are there papers on it that I can read?

Update: Question has some answers here: Continuum between addition, multiplication and exponentiation?
here: Example $x$, $y$ and $z$ values for $x\uparrow^\alpha y=z$ where $\alpha\in \Bbb R-\Bbb N$
and here: https://www.hindawi.com/journals/mpe/2016/4356371/#abstract

Michael Smith
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  • For non-negative integer values of the order, this is called hyperoperation. So you can try searching for things like "non-integer hyperoperation" or "fractional hyperoperation". E.g. see https://math.stackexchange.com/questions/1350025/hyperoperation-sequence-with-non-integer-values-of-n or https://math.stackexchange.com/questions/1227761/example-x-y-and-z-values-for-x-uparrow-alpha-y-z-where-alpha-in-bbb/1241979#= – Minus One-Twelfth May 17 '20 at 01:10
  • Thanks a ton for those links. This is super interesting stuff. From what I've read so far, it appears there's not an accepted way to find fractional, or imaginary ranks of hyper-operations. – Michael Smith May 17 '20 at 01:26
  • I don't see how this is related to [complex-analysis]. Perhaps you meant [computational-complexity]? – Martin R May 17 '20 at 05:25
  • I was under the impression that real and complex analysis were the study of expanding functions beyond their natural scope. Such as the Gamma function as an expansion of factorials, etc. Forgive me if my tags are not correct. Do I have the wrong impression? – Michael Smith May 17 '20 at 07:01

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