I was reading about tetration [here](http://www.tetration.org/Tetration/index.html). The site mentions that the convergence of the expansion for fractional iteration is unproven. However, I was interested in reading more literature about convergence in special cases perhaps.


In addition, is there an explicit form for the series expansion? I'd like some references for that too. The site only lists a couple of terms.


(I'm not associated with any institutions so public access is best)



Convergence of expansion for fractional iteration
**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](http://www.tetration.org), there is to my knowledge no prior work to reference.


**Mathematica**


[Code for hyperbolic fractionaly iterated functions](http://www.tetration.org/Dynamics/DynDerv/index.html)


**Complex Ackermann Function**


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


