[FOM] Fractional Iteration and Rates of Growth

[Dmytro Taranovsky]

Joe Shipman wrote:
>If, as you say, there is a unique real analytic function g(x) = 
>f(1,x,0) defined by analytic continuation along the (positive) real 
>axis, such that g(1)=1, g(2)=2, g(3)=4, g(4)=16, g(5)=65536 
>[satisfying g(x+1)=2^g(x)]

I did not say that.  Instead, we have
f(n, x, 0) = 0 (n>0)
If g(x) = x*2^x, then 
f(2, x, 1) = gg...1 (g is applied x times)
f(3, 1, x) = gg...x (g is applied x times)

(Recall that f(0, a, x)=x+a, f(n+1, 1, x) = f(n, x, x), and f(n, a, x)
is the ath iterate (in the appropriate sense) of f(n, 1, x).)

>Can you
>1) Prove that g and all its derivatives are monotonically increasing?

If an analytic g is defined for all positive real numbers and with all
derivatives non-negative (at some particular non-negative real number),
then g is entire, so the above requirement is too strong.
However, f(2, x, 1) should be defined for all real x, and I do not know
whether it satisfies the above condition.

>2) Prove that g satisfies the functional equation g(x+1)=2^g(x) 
>everywhere it is defined? (Obviously its domain cannot be extended 
>to negative integers if this is the case.)

f does satisfy an appropriate functional equation (at least if the
analytic continuation is done along the real line only).

>3) Calculate g(3.5) and g(4.5) to the nearest integer?

I expect that for every n, f is primitive recursive (as a function from
real numbers to real numbers).

>4) Specify a maximal subdomain of the complex plane to which f 
> can be extended?

Not yet. I still do not know whether f(n, a, x) is defined for all
positive x and real a.

Dmytro Taranovsky