11/22/2015, 08:12 PM
I found the thread that mentions log(x^x^x) in it: http://math.eretrandre.org/tetrationforu...hp?tid=280
Superroots and a generalization for the LambertW

11/22/2015, 08:12 PM
I found the thread that mentions log(x^x^x) in it: http://math.eretrandre.org/tetrationforu...hp?tid=280
11/24/2015, 12:51 AM
I believe I may have found a closed form for the third superroot / generalized LambertW function:
Regards, Andrew Robbins (11/24/2015, 12:51 AM)andydude Wrote: I believe I may have found a closed form for the third superroot / generalized LambertW function: Hah, that sounds good, I'll try it tomorrow! (I've just seen formulae 96100 in your earlier announced paper, but can read it also not before tomorrow afternoon) Did you see already whether it is possibly simply extensible to higher orders? Gottfried
Gottfried Helms, Kassel
11/24/2015, 07:00 AM
I believe I found a slightly smaller or cleaner closedform for the above function
Regards, Andrew Robbins
11/24/2015, 07:16 AM
(11/24/2015, 02:56 AM)Gottfried Wrote: Did you see already whether it is possibly simply extensible to higher orders? I tried doing something similar with superroot4, but no luck, however, I found the coefficients by solving the equation so you should be able to add to the above formula to solve this generalization. The above formula is just the special case when .
[text updated]
Having not yet studied Andrew's formulae, I just played around with the idea of iterated superroots. In this case, instead of with x being the second, third superroot of some y, I reconsidered simply the iterated second superroot  which is easier to implement, because and for LW (the LambertW) there are easy implementations in M'tica and Pari/GP. Of course, for we have, that  where x is a bit smaller than (of course the latter is what I tried to approximate by some iteration). Now I found the following amazing procedure. Consider for example . Then compute up to some limit. Then the inhomogenuous exponentialtower / "nested exponentiation" (wikipedia) . Of course, a bit thinking about this makes it clear that this is a nearly trivial matter; but the amazing part of it is, to get a new intuition for a general/nested exponential tower, where the single stairs are not equal but follow some functional description... and, for instance, might be interpolated to give some fractional interpolation of the H2()procedure ... In Pari/GP: Code: h2(x)= exp(LW(log(x))) \\ define h2(x), use LambertWimplementation Nice... Here the vectorv vx (read from left to right, then top down): Code: 11.9551115478 2.59837825984 1.73428401000 1.45860478939 Appendix: the code for LambertW, taken from wikipedia: Code: LW(x, prec=1E80, maxiters=200) = local(w, we, w1e); For y=27^27 we get the following vector vx of "stairs": Code: 27.0000000000 3.00000000000 1.82545502292 1.49546396135 y = 27^27 = 27 ^ 3 ^ 1.82... ^ 1.49... ^...
Gottfried Helms, Kassel
12/01/2015, 11:58 PM
@andrew
Congrats with your result. @gottfried The thing is solving (x_m ^ x_m)^[m] = y is only close to solving X_n^^[n] = y ( n = m in value ) When Y is large and n (or m) is small. For instance x in x^x^x^x = 2000 is close to Y in (y^y)^(y^y) = 2000. But a in a^a^a^a = 2,718 is different from B in (b^b)^(b^b) = 2,718. This is logical considering the fixpoint X^x = x Gives x = {1,1}. So one method is attracted to eta and the other to 1. For y > e that is. For y < e its even worse. Since we are mainly intrested in small y and Large n ... This idea seems not so practical here. Guess it might be more usefull for the basechange .... Well Maybe ... Regards Tommy1729
12/02/2015, 12:48 AM
@Tommy1729
Thanks! @Gottfried By "inhomogenuous" do you mean "heterogeneous"? Iterated superroots? I need some time to wrap my head around this...
12/02/2015, 02:43 AM
Hi Andrew 
(12/02/2015, 12:48 AM)andydude Wrote: By "inhomogenuous" do you mean "heterogeneous"? Well, this might also be correct. I simply mean that the powertower has varying entries (tetration has one fixed entry, the base, except the top one which if equals 1 can be omitted). In wikipedia it is proposed to call it "nested exponentiation" (I forgot that). Quote:Iterated superroots? I need some time to wrap my head around this...I think, Henryk had discussed them in his dissertation? (I'm not sure). Just Gottfried
Gottfried Helms, Kassel
(12/01/2015, 11:58 PM)tommy1729 Wrote: The thing is solving (x_m ^ x_m)^[m] = y is only close to solvingTrue. But having this way a (nontrivial) vector of different exponents (or better: bases) which comes out to be a meaningful "nested exponentiation" I'm curious, whether one can do something with it, for instance weighting, averaging, or multisecting that sequence of exponents/bases when recombining them to a "nested exponential". We have not yet many examples of "nested exponentiations" with a meaningful outcome. For instance, the construction of the Schroederfunction is based on (ideally) infinite iteration of the basefunction to get a linearization. If we iterate the h2()function infinitely, the curve of the consecutive values in an x/ydiagram (where x is the iteration number) approach a horizontal line; don't know whether using that linearization shall prove useful for something similar. (When Euler found his version of the gammafunction, that was in one version putting together sequences of integer numbers weighting and repeating in a meaningful way; there is some infinite productrepresentation for his gammafunction I think I recall correctly... ) (see also the updates in my previous (introducing) posting)
Gottfried Helms, Kassel

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