Re: Tetration again!

On Dec 21, 1:29 pm, mike3 <mike4...@xxxxxxxxx> wrote:

Hmm. Does this mean the true solution of tetration
was already discovered, a real long time ago?

there isn't really a true solution to a process
there are only properties
identities
various ways of characterising and using the process

there is quite a history to iterated exponentiation
but there is much still unknown

even just bell numbers
still likely have a number of identities waiting
considering all their combinatorial significance

there is much still to understand
about the general combinatorics of the iterated case

gottfried and others have done work on series
like the alternating sum he has posted about

what can we say about general

---
\ i
/ a x
--- i

and other related forms?

convergence criteria?

it has always been a very interesting world
because it has a lot of structure
but is complex enough not to have really easy properties

there is quite a history
though

much has been discovered of the dynamics of the iteration
topologically
stability
..

euler
eisenstein
and others gave some of the first descriptions

and many names have followed

seidel
thron
rota
schubert
...

knowing what to tackle
requires following much of the history

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.

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