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.
\( x{\uparrow}{\uparrow}n = \sum_{k=0}^{\infty} p_{nk} \ln(x)^k \) 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: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:
