**Convergence of Fractional Iterration**

I looked into the literature on the convergence of fractional iteration and had copied some resent works, but I found I needed a number of supporting papers I had no access to in order to master the papers in my possession. A single source covering the topic would be a great service.

**Explicit Series Expansion**

Let $f(z)$ and $g(z)$ be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.

$D^nf(g(z))=$
$\sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} $
$(D^kf)(g(z))$
$\left(\frac{Dg(z)}{1!}\right)^{k_1} $
$ \cdots \left(\frac{D^ng(z)}{n!}\right)^{k_n}$

A partition of $n$ is $\pi(n)$, usually denoted by $1^{k_1}2^{k_2}\cdots n^{k_n}$ with $k_1+2k_2+ \cdots nk_n=k$; where $k_i$ is the number of parts of size $i$. The partition function $p(n)$ is a decategorized version of $\pi(n)$, the function $\pi(n)$ enumerates the integer partitions of $n$, while $p(n)$ is the cardinality of the enumeration of $\pi(n)$.

Setting $g(z) = f^{t-1}(z)$ results in

$D^n f^t(z) = \sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} $
$(D^k f)(f^{t-1}(z))$
$\left(\frac{Df^{t-1}(z)}{1!}\right)^{k_1} $
$ \cdots \left(\frac{D^n f^{t-1}(z)}{n!}\right)^{k_n}$

The Taylors series of $f^t(z)$ is derived by evaluating
the derivatives of the iterated function at a fixed point
$f^t(0)$ by setting $z=0$ and separating out the $k_n$
term of the summation that is dependent on $D^n f^{t-1}(0)$.

$D^n f^t(0) = \sum \frac{n!(D^k f)(0)}{k_1! \cdots k_{n-1}!} $
$\left(\frac{Df^{t-1}(0)}{1!}\right)^{k_1} \cdots $
$\left(\frac{D^n f^{t-1}(0)}{(n-1)!}\right)^{k_{n-1}} $
$ + (D f)(0) D^n f^{t-1}(0)$

The remaining $p(n)-1$ terms of the summation are only dependent on $D^k f^{t-1}(0)$, where $0<k<n$.

**References**

As the author of the mentioned tetration website, there is to my knowledge no prior work to reference.

**Mathematica**

Code for hyperbolic fractionaly iterated functions

**Complex Ackermann Function**

The series expansion of iterated functions provides a way to define complex tetration and even the complex Ackermann function.