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 The Power Series definition for tetration beboe Newbie Posts: 1 Threads: 1 Joined: Sep 2008 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 ? can X tet X be decribed using product series? andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 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:     Gottfried Ultimate Fellow Posts: 764 Threads: 118 Joined: Aug 2007 09/14/2008, 10:03 AM 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 Gottfried Helms, Kassel bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 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 ReferenceAndrew wrote down there also the direct series development at (fixed point) x=1. Gottfried Ultimate Fellow Posts: 764 Threads: 118 Joined: Aug 2007 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... Gottfried Attached Files Image(s)     Gottfried Helms, Kassel « Next Oldest | Next Newest »

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