[FOM] 605: Integer and Real Functions (Harvey Friedman)

[Timothy Y. Chow]

Harvey Friedman wrote:
> So what we would like to have is a TWO VARIABLE functional equation with 
> a initial condition that uniquely determines factorial on R|>=0.

One thing to try, since you have a two-variable functional equation for 
exponentiation, is to use the fact that the gamma function is the Mellin 
transform of exp(-x).  Of course then one has to come up with a 
"fundamental purpose" for the Mellin transform.  Perhaps the fact about 
the Mellin transform that is closest in spirit to what you are trying to 
do here is that if X and Y are two independent random variables, then the 
Mellin transform of their products is equal to the product of the Mellin 
transforms of X and Y.

> QUESTION: Find some natural classes of functions, X, such that for all
> f in X there is a (unique) g in X such that f = gog.
>
> Horrible things seem to happen when you look at, e.g.,
>
> 2^n = g(g(n)) on N
> e^x = h(h(x)) on R|>=n.
>
> The intuitive idea is clear. Let f(x) be the result of doing something
> to x. Then g(x) should be doing it half way. When, there, and how does
> this make sense?

Well, there's a lot of literature on this subject, e.g.,

http://reglos.de/lars/ffx.html

I'm not sure that there's anything that you'd find very satisfying, 
though.  For example, there do exist moderately well-behaved solutions to 
e^x = h(h(x)) but none that are holomorphic, so it's not clear that any of 
the solutions are "fundamental."

Tim