A tiny base-dependent formula for tetration (related to "change-of-base"?)
Well, the following is not yet deeply considered; it is more that I stumbled across this observation, and did not yet really analyze all its implication but want to announce, so someone else may crosscheck and look at it (I've my head elsewhere currently)
When rereading my article about "pascalmatrix tetrated" (*1) and playing/checking some computations, it came to my attention, that I already have a formula for sexp with fixed (integer) height parameter, but variable with the base. So feeding log(2) as parameter to the series may give 2^^3, and feeding log(3) gives then 3^^3 . In all my recent consideration the base-parameter was deeply calculated in the coefficients and the height-parameter was isolated and thus subject to change; here the height-parameter is hidden in the coefficients and the base-parameter is explicit and can be subject to change.
We need simply the first column of the tetrated pascalmatrix, scale it by reciprocal factorials and use it as coefficients for the powerseries:
\( {b\^\^}^h = \sum_{k=0}^{\infty} \frac{{P\^\^}^h_{k,0}}{k!}\log(b)^k \)
where P^^h is the tetrated pascalmatrix as described in the article with a given integer height h, and the indexes in the formula denote the k'th rows in first column.
(Clearly this has range of convergence and such, just to be determined soon.)
Gottfried
(*1) http://go.helms-net.de/math/tetdocs/Pasc...trated.pdf
The fact of representing an exponential generating-function was also given in the OEIS but it didn't appear to me, that this is (thus) a base-dependent formula; for instance see for h=2:
(*2) http://www.research.att.com/~njas/sequences/A000248
Well, the following is not yet deeply considered; it is more that I stumbled across this observation, and did not yet really analyze all its implication but want to announce, so someone else may crosscheck and look at it (I've my head elsewhere currently)
When rereading my article about "pascalmatrix tetrated" (*1) and playing/checking some computations, it came to my attention, that I already have a formula for sexp with fixed (integer) height parameter, but variable with the base. So feeding log(2) as parameter to the series may give 2^^3, and feeding log(3) gives then 3^^3 . In all my recent consideration the base-parameter was deeply calculated in the coefficients and the height-parameter was isolated and thus subject to change; here the height-parameter is hidden in the coefficients and the base-parameter is explicit and can be subject to change.
We need simply the first column of the tetrated pascalmatrix, scale it by reciprocal factorials and use it as coefficients for the powerseries:
\( {b\^\^}^h = \sum_{k=0}^{\infty} \frac{{P\^\^}^h_{k,0}}{k!}\log(b)^k \)
where P^^h is the tetrated pascalmatrix as described in the article with a given integer height h, and the indexes in the formula denote the k'th rows in first column.
(Clearly this has range of convergence and such, just to be determined soon.)
Gottfried
(*1) http://go.helms-net.de/math/tetdocs/Pasc...trated.pdf
The fact of representing an exponential generating-function was also given in the OEIS but it didn't appear to me, that this is (thus) a base-dependent formula; for instance see for h=2:
(*2) http://www.research.att.com/~njas/sequences/A000248
Code:
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FORMULA E.g.f.: exp(x*exp(x)).
G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 25 2003
Gottfried Helms, Kassel
