Re: Tetration again!

On Dec 20, 5:43 am, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 20.12.2007 08:36 schrieb galathaea:

On Dec 19, 10:00 pm, lwal...@xxxxxxxxx wrote:

I believe the reason Gottfried uses U instead of the usual
tetration, which he calls T, because the function to iterate,
namely b^x - 1, has an obvious fixed point, x = 0, and its much
easier to write a Taylor series centered at a fixed point. The
usual function e^x, for example, has only complex fixed points.

U has many very nice properties
that make its study "natural"

it was first studied deeply by ramanujan
(in a slightly different form)
and was one of the first successfully tackled form
for iteration of exponentials

later this was taken up by bell
and the first interesting power series

oo
x ---
e - 1 \ B(i) i
e = / ---- x
--- i!
i=0

define what are now known as the bell numbers

focussing just on the ramanujan iterates defined by

R_m(x)
R (x) = e - 1
m+1

relations can be derived for derivatives easily

eg.
n
---
R'(x) = | | (1 + R (x))
n | | i
i=1

and so on

many of those who have worked on iterated exponentials
(bell, carlitz, ..)
turn to this form for computational ease
looking at transformations to get results on other forms

cf. ramanujan's chapter 4
riordan
or comtet

Ah, that is a nice hint. Could you provide more info
about the ramanujan-source? (to be able to look up in
the lib or even online)

ramanujan's notebooks are available
in two forms i am aware of

one was issued by the tata institute
just as a photocopy
the other is a nicely edited one by bruce berndt

the berndt editing has known proofs
and other very useful exposition
separated into 5 books

chapter 4 of ramanujan's notebooks
is found in book 1 of berndt
which he titles:
"iterates of the exponential function
and an ingenious formal technique"

entries 1 thru 7 (and their corollaries)
give his basic results
including basic recursions
series identities
an integral formula
and (as always for ramanujan) asymptotics

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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