(09/26/2009, 09:25 PM)bo198214 Wrote: In response to the thread Solving tetration for base 0 < b < e^-e. I will feature here what one could call regular superlogarithm.
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And then we consider the first line of the matrix power \( C^t \). These are the coefficients of \( f^{\circ t} \):
\( 0\\
\left(-1\right)^{t}\\
-\frac{1}{4} \, \left(-1\right)^{t} + \frac{1}{4}\\
\frac{1}{12} \, t \left(-1\right)^{{(t - 1)}} + \frac{1}{8} \, \left(-1\right)^{t} - \frac{1}{8}\\
-\frac{1}{16} \, t \left(-1\right)^{{(t - 1)}} - \frac{7}{96} \, \left(-1\right)^{t} - \frac{1}{24} \, t + \frac{7}{96}\\
\frac{1}{96} \, {(t - 1)} t \left(-1\right)^{{(t - 2)}} + \frac{7}{160} \, t \left(-1\right)^{{(t - 1)}} + \frac{3}{64} \, \left(-1\right)^{t} + \frac{1}{24} \, t - \frac{3}{64}\\
-\frac{5}{384} \, {(t - 1)} t \left(-1\right)^{{(t - 2)}} + \frac{1}{144} \, {(t - 1)} t - \frac{1}{32} \, t \left(-1\right)^{{(t - 1)}} - \frac{41}{1280} \, \left(-1\right)^{t} - \frac{181}{5760} \, t + \frac{41}{1280}\\
\frac{5}{3456} \, {(t - 2)} {(t - 1)} t \left(-1\right)^{{(t - 3)}} + \frac{1}{90} \, {(t - 1)} t \left(-1\right)^{{(t - 2)}} - \frac{1}{96} \, {(t - 1)} t + \frac{365}{16128} \, t \left(-1\right)^{{(t - 1)}} + \frac{263}{11520} \, \left(-1\right)^{t} + \frac{263}{11520} \, t - \frac{263}{11520}\\
\dots
\)
Each coefficient is a mixture of polynomials in \( t \) containing \( (-1)^t \). It has 0 convergence radius.
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Hi Henryk -
I was curious how you derived that coefficients but am unable to apply your definitions using integral and Julia-function from the beginning of your post.
Unfortunately I have written my own interpretation of the coefficients for the iterates of exp(x)-1 in a form, where (u-1),(u^2-1) etc occur in the denominators and thus cannot easily be evaluated for -for instance- u=-1 which is the case you're discussing here. I derived them using the symbolic eigensystem-decomposition and have documented them here
http: go.helms-net.de/math/tetdocs/APT.htm
On the other hand, there is also a method to derive them directly from the list of coefficients for increasing powers using a special scheme of weighted forward differences. This all seems finally to be connected if expressed in terms of q-analogues, and such a formulation seems then to allow to express the coefficients for any base-logarithm "u", including u=1, u=-1, and all complex unit-roots - using the matrix-method I had to choose between the cases u=1 (mat-log) |u|<>1 (diagonalization) and |u|=1,u=/=1 (impossible to express) so far.
I see, how your polynomials can be used to compute the coefficients for any power, the results agree with the list of coefficients as they occur for the integer iterates of exp(-x)-1, so they must be correct, but seem to be compressed telling only the sums of the numerically indistinguishable powers of (-1)
To make things waterproof for me I'd like to see your formula (or better the list of coefficients) in generalization , where the variable u is inserted for the parameter (-1) of your current example. So that for u=log(2) we can generate the coefficients for the t'th iterate of 2^x-1 ,for u=1 we can generate the coefficients of the t'th iterate of exp(x)-1 and for u=-1 we can generate your coefficients for the t'th iterate of exp(-x)-1.
Does your computation-scheme allow to insert the general expression "u" (I am used to "u" for the log of the base for the dxp-function )?
Gottfried
P.s. My own new expression is very simple.
define the bell-matrix Ut for exp(u*x) - 1 (keeping "u" as symbol), then generate a list of the coefficients for the k'th iterated function using the k'th integer power of Ut and extract its second columns into the k'th column of the list.
Then define the triangular matrix of ones, scaled by inverse powers of u:
Code:
DR =
1 . . . .
1 1/u . . .
1 1/u 1/u^2 . .
1 1/u 1/u^2 1/u^3 .
1 1/u 1/u^2 1/u^3 1/u^4
...DRI = DR^-1
Then, using k as index for the column in list and also for the iteration-index for exp(u*x)-1 ,
DRI^k*list[,k] = column(a0,a1,...,ak, 0,0,0,0,...)
reduces the coefficients to the sequence of zeros with some parameters a0,a1,...ak at the beginning and thus define a finite recursion for the computation of coefficients using the (powers of) DR with that parameters as initial values.
Here, DRI consists of the q-binomials to base -1 .
While I could reproduce some examples from your coefficients by a q&d-check with that procedure I did not yet set up a clean description, I'll come up with it later, perhaps wednesday or friday.
Gottfried
Gottfried Helms, Kassel