05/21/2008, 06:24 PM

Inspired by Kouznetsov's consideration I investigated a bit more in this direction and found interesting results:

1.

Let F be holomorphic on the right half plane, and let F(z+1)=zF(z),

F(1)=1, further let F be bounded on the strip 1<=Re(z)<2, then F is

the gamma function.

2.

is also determined as the only function that satisfies , and is bounded on the strip 1<=R(z)<2 (or 0<=R(z)<1).

3. I started a thread in sci.math.research and it turned out that the same criterion also makes the Fibonacci function

, the unique extension of , , .

So I really would guess that this criterion (which is a slightly weaker demand than Kouznetsov's criterion) also implies uniqueness for tetration, at least for base .

Maybe its not even difficult to prove.

1.

Let F be holomorphic on the right half plane, and let F(z+1)=zF(z),

F(1)=1, further let F be bounded on the strip 1<=Re(z)<2, then F is

the gamma function.

2.

is also determined as the only function that satisfies , and is bounded on the strip 1<=R(z)<2 (or 0<=R(z)<1).

3. I started a thread in sci.math.research and it turned out that the same criterion also makes the Fibonacci function

, the unique extension of , , .

So I really would guess that this criterion (which is a slightly weaker demand than Kouznetsov's criterion) also implies uniqueness for tetration, at least for base .

Maybe its not even difficult to prove.