09/14/2008, 03:51 AM (This post was last modified: 09/14/2008, 09:44 AM by andydude.)
What would be the Power Series for X tetrated to the 2nd, X tetrated to the 3rd, 4th, ...etc. Then find a pattern in these series for a general series definition of X Tetrated to any X.
The Series for:
X tet 2 = X ^ X = exp ( X ln X )=
Sigma ((X ln X) ^ n) / n!
for n = 0 to infinity.
now let this series = A
Then
the series for X tet 3 = X ^ A = exp (A ln X) =
Sigma ((A ln X) ^ m)) / m!
for m = 0 to infinity
now let this series = B
Then continue the process for more higher nested series...
Can anyone express the nested series for the tetration powers of 2 and higher as just a single power series?
IF this can be shown, is their any pattern to these Sigma expressions to give a generalized power series ?
09/14/2008, 08:13 AM (This post was last modified: 09/14/2008, 08:16 AM by andydude.)
These are what I call Puiseux series of tetrate functions. They were first discussed in detail by Galidakis (in this paper, see also this page). I call them Puiseux series because according to MathWorld, they're series involving logarithms.
beboe Wrote:Can anyone express the nested series for the tetration powers of 2 and higher as just a single power series?
Ioannis Galidakis can. He gave this recurrence equation in his paper:
\( x{\uparrow}{\uparrow}n = \sum_{k=0}^{\infty} p_{nk} \ln(x)^k \) where \( p_{nk} = \frac{1}{k} \sum_{j=1}^{k} j p_{n(k-j)} p_{(n-1)(j-1)} \)
for more information, please see section 4.2.3 (page 26) in the Tetration Reference
beboe Wrote:IF this can be shown, is their any pattern to these Sigma expressions to give a generalized power series ?
If only it were that simple...
beboe Wrote:can X tet X be decribed using product series?[/b]
I don't know... but I think it kinda looks like this:
beboe Wrote:What would be the Power Series for X tetrated to the 2nd, X tetrated to the 3rd, 4th, ...etc. Then find a pattern in these series for a general series definition of X Tetrated to any X.
The Series for:
X tet 2 = X ^ X = exp ( X ln X )=
Sigma ((X ln X) ^ n) / n!
for n = 0 to infinity.
now let this series = A
Then
the series for X tet 3 = X ^ A = exp (A ln X) =
Sigma ((A ln X) ^ m)) / m!
for m = 0 to infinity
now let this series = B
Then continue the process for more higher nested series...
Can anyone express the nested series for the tetration powers of 2 and higher as just a single power series?
IF this can be shown, is their any pattern to these Sigma expressions to give a generalized power series ?
can X tet X be decribed using product series?
Hmm, what prevents you, to just to try this, and show us what you get for tet 2, tet 3... ?
For decremented exponentiation you may find this article interesting. powerseries iteration
09/14/2008, 01:58 PM (This post was last modified: 09/14/2008, 02:01 PM by bo198214.)
andydude Wrote:
beboe Wrote:Can anyone express the nested series for the tetration powers of 2 and higher as just a single power series?
Ioannis Galidakis can. He gave this recurrence equation in his paper:
\( x{\uparrow}{\uparrow}n = \sum_{k=0}^{\infty} p_{nk} \ln(x)^k \) where \( p_{nk} = \frac{1}{k} \sum_{j=1}^{k} j p_{n(k-j)} p_{(n-1)(j-1)} \)
for more information, please see section 4.2.3 (page 26) in the Tetration Reference
Andrew wrote down there also the direct series development at (fixed point) x=1.
09/24/2008, 06:29 PM (This post was last modified: 09/24/2008, 08:26 PM by Gottfried.)
"Just derived a method to compute exact entries for powers of the matrix-operator for T-tetration.
(...)"
[deletion] I moved my reply to the "matrix-method"-thread since it didn't reflect, that the OP-question asked for a powerseries in the height-variable while I discussed one in terms of log of the base-parameter and the top-parameter x. Sorry for messing things...
