Can Hyperexponents be Extended to Non-integers?

Is it possible to have a hyperexponent that is not a whole positive number? What does it mean to have a negative hyperexponent? What does it mean to have a fractional one? In standard exponents, x^y means that x is multiplied by itself y times. When y is not a whole number, assuming y is rational, it can be broken down into a fraction, say a/b, then you take the bth root of x^a. Can these ideas be extended to hyperexponents?
- Ray (age very old)
Champaign, IL, US

Ray- Wow that's a tough one. I tried playing with it a little and got no obvious analytic extension of the concept to non-integers or negatives. So I fell back on Wikipedia:. It turns out that it was a good idea not to try to spend too much more effort. Apparently there is no unique solution to the extension of the integer hyperexponential to non-integer real values. However, the article says there's an unproven conjecture that the extension to complex powers is unique, if required to be analytic (holomorphic) everywhere except for real values less than or equal to -2. So far as I can tell, there aren't any nice simplifications like those you mention for ordinary exponents.

Just for your amusement, here's a relatively easy hyperexponent problem:

Let x^^infinity=2, where ^^ is the hyperexponent operator.  What's x?

A little harder:
Let x^^infinity=y. What's the largest positive real y for which that has a solution?

Mike W.

(published on 02/07/2011)