I was reading about tetration here. 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 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.
This is a very large and old subject. Here are some important papers of fractional iteration:
Écalle, J. Théorie itérative: introduction à la théorie des invariants holomorphes. (French) J. Math. Pures Appl. (9) 54 (1975), 183–258.
S. Voronin. Analytic classification of germs of conformal mappings (C, 0) → (C, 0). Funktsional. Anal. i Prilozhen 15 (1981), no. 1, 1–17, 96 (in Russian). English translation: Functional Anal. Appl. 15 (1981), no. 1, 1–13.
Baker, I. N. Permutable power series and regular iteration. J. Austral. Math. Soc. 2 1961/1962 265–294.
A further -likely useful- reference is L. Comtet, Advanced combinatorics; in the 1974 edition from page 137 he shows the function composition using the Bell-polynomials and from page 144 he even provides the Fáa di Bruno-formula in a more concise matrix-notation (Comtet references here to earlier work of E. Jabotinsky and others); from there he arrives at a formula for the coefficients of the fractional iterate of $f(x)$. The functions in question here are of the form $ \displaystyle f(x) = \small \sum_{k=1}^\infty a_k{ x^k \over k!}$ and that of the $h$'th iterate $ \displaystyle f^{\circ h}(x) = \small \sum_{k=1}^\infty A_k(h){ x^k \over k!}$ where $ \small A_k(h)$ are polynomials in $h$.
The matrix-machinery given here reflects the ideas of E. Schroeder (of about 1870 I think) for fractional iterates of a (suitable) function and leads to the problem of diagonalization of infinite triangular matrices (see "Bell-matrix", "Carleman-matrix") when in $f(x)$ the coefficient $a_1 \gt 0 , a_1\ne 1$ and to the problem of matrix-logarithm of the Carleman-matrix, when $a_1=1$ .
For the case $f_t(x) = t^x - 1 $ which has a power series expansion $f_t(x) = ux + {(ux)^2 \over 2! } + {(ux)^3 \over 3! } + ... $ (writing "$u$" for "$\log(t)$" ) I managed to expand the polynomials $A_k(h,u)$ into a nice matrix-arrangement of constant integer numbers independent of $u,h$ (containing Striling numbers first and second kind at the edges) but could not yet find a formula to produce the $A_k(h,u)$ directly instead of the naive recursive computation which occurs when one solves for the diagonalization of the triangular Carlemanmatrix for $f_t(x)$. (see an example in my tetration-pages , scroll down some screen-pages).
Now having a definition for the fractional iterates of, say, $f_t(x)=t^x-1$ is not really tetration, where we want fractionally iterate, say, $g_b(x)=b^x$. But a solution for $f_t(x)$ can be related to a problem $g_b(x)$ by a recentering of the power series to a fixpoint. We can write the formal power series $$ g^{\circ h}_b(x) = \lambda_b + f^{\circ h}_t(x-\lambda) $$ when $b=t^{1 \over t} = \exp( u \exp(-u))$; so for instance if $b=\sqrt 2$ then $t=2$ and $u= \log 2 = 0.693... $ . The coefficient $\lambda_b$ is here a fixpoint for $g_b(x)$ and is identical with $t$ in this case. However, $t$ is not unique, the equation $b=t^{1 \over t}$ has in general multiple solutions for $t$. And then the fractional iterates, taken by that mechanism, are different for the different $t$ !
For the fractional iteration of $f_e(x) = \exp(x)-1$ I.N. Baker has proven, that the occuring power series are all divergent if the iteration-height parameter $h$ is noninteger. I don't know at the top of my head a reference to the equivalent statement for $f_t(x)$ with a general parameter $t$, but in all my experiences with general $t$ the series had strong divergence in their coefficients such that the radius of convergence of the series should always be zero. See an illustration for the power series for fractional heights of $f_e(x)=\exp(x)-1$ at my pages here, here or here . Perhaps the best/most complete reference is this
Remark: In the above I've left out all the restrictions on $b$, $t$ etc which are required to get series in real numbers etc; I think this sloppyness is compensated by a more readable exposition of the general problem
Hoping that it could be helpful, let us follow the linked literature to the OEIS sequence $A35172$ (see https://oeis.org/A351727). Here, we should be able to get some very useful references, as Paulsen's valuable works "Tetration for integer bases" (2019) and also the noteworthy discussion "new fatou.gp program" on the Tetration Forum. Complex tetration is a fascinating open field and I found Paulsen's research very intriguing.
