Fractional calculus and tetration

[sheldonison]

Well hello everybody. It's been a while since I posted here. I've been working vivaciously on iteration and fractional calculus and the ways the two intertwine and I've found a nice fact about tetration.

I've been able to prove an analytic continuation of tetration for bases \( 1 < \alpha < e^{1/e} \) and I've been wondering about the base change formula, if this admits a solution for \( \alpha > e^{1/e} \). The solution I've generated is periodic with period \( 2\pi i / \log(\beta) \) where \( \beta \) is the attracting fixed point of \( 1 < \alpha < e^{1/e} \)....

I just posted a similar comment on Mathstack. I like it enough to copy it here...

Technically, \( ^x b \), for \( b<e^{1/e} \), The Kneser Tetration becomes ambiguous. For real bases \( b>e^{1/e} \), Tetration is well defined, and analytic with singularities at negative integers<=-2. The base \( b=e^{1/e} \) is the branch point, where iterating no longer grows arbitrarily large. I investigated what happens to Tetration when we extend it analytically to complex bases, and it turns out that for \(b<e^{1/e}\), Tetration is no longer real valued at the real axis. See http://math.eretrandre.org/tetrationforu...hp?tid=729

Consider \( b=\sqrt{2} \), which has two fixed points, L1=2, and L2=4. Most of the time, when people talk about Tetration for \( 1<b<e^{1/e} \), they switch to looking at the attracting fixed point, in this case L1=2. Then Tet(z) has the familiar definition, logarithmic singularity at Tet(-2), Tet(-1)=0, Tet(0)=1, and Tet(1)=b, and in the limit as \( n\to \infty \), you get the attracting fixed point L1, which is 2 for \( b=\sqrt{2} \). But for bases \( b>e^{1/e} \), we are using both complex conjugate fixed points to generate Kneser's real valued at the real axis Tetration. And if we move the base in a circle around \( e^{1/e} \) slowly using complex bases, from a real base greater than \( e^{1/e} \) to one less than \( e^{1/e} \), then we get to a function still uses both the attracting and repelling fixed points, but the function is no longer real valued at the real axis. Using the attracting fixed point is not the same function as Kneser's Tetration.

Ok, now about the "base change" function. I could find a link on this forum, but here is a very short description: If you develop real valued Tetration by iterating the logarithm of another super-exponentially growing function, you get a function that is infinitely differentiable and looks a lot like Tetration, but it turns out to be nowhere analytic. Lets say \( f(x) \) is a super-exponentially growing function, and we want to develop the "base change" Tetration for \( b>e^{1/e} \) by iterating the logarithm of f(x) as follows:

\( \text{Tet}_b(x) \; = \lim_{n \to \infty} \log_b^{o n} \left(f(x+k_n+n)\right); \;\;\;\;\; f(k_n+n)\; = \; ^n b; \;\;\;\; k_n\; \) quickly converges to a constant as n increases

If f is Tetration for another base, or many other super-exponentially growing functions, then it turns out that all of the derivatives at the real axis converge, but they eventually grow too fast for this base change Tetration function to be an analytic function. Also, the base change function is not defined in the complex plane. I haven't posted a rigorous proof of the nowhere analytic result.