(06/12/2022, 06:17 AM)Daniel Wrote: See tetration combinatorics for more information. The example derives \( D^4f^n(z) \). The following works for all smooth iterated functions, so it applies to tetration and the hyperoperators.
Enumerate the total partition of 65536 (not computationally practical), evaluate each enumeration and add the terms together.
Hey, Daniel. Catullus was asking about Kneser. This does not solve Kneser. This only solves the geometric solution about a Fixed point (Schroder iteration).
To Catullus:
For Shell-Thron region. I can calculate arbitrary Taylor series of \(b\uparrow \uparrow z\) of \(z\) about any point \(z_0\). I can do this using the beta method (which solves for varying periodic solutions) and for the standard Schroder solution. Though I've never released code from the Schroder equation, I can do it. Then if you ask for the taylor series with 500 terms about 3, set the series precision to 500, then you'd just run sexp(3+z) for whatever \(b\) is. But this only works within the Shell-thron region. And it is much slower than fatou.gp. Be prepared to wait. But it will produce the taylor expansion to desired requirement.
The beta method will do this faster, but it will produce a different taylor series which is \(2\pi i / \lambda\) periodic, where \(\lambda\) is now a free variable as opposed to being fixed like with Schroder. And the Schroder iteration exists as a boundary value of the beta method. So you can't use the beta method to get Schroder unfortunately.
