Superroots and a generalization for the Lambert-W

[andydude]

I found the thread that mentions log(x^x^x) in it: http://math.eretrandre.org/tetrationforu...hp?tid=280

[andydude]

I believe I may have found a closed form for the third superroot / generalized LambertW function:
\(
{}^{3}W(v) = \log\left(\sqrt[3]{e^v}_s\right) = \sum_{k=0}^{\infty}
\frac{v^k}{k!} \sum_{j=0}^k
{k-1 \choose j}j(k-j)^{j-2}(-k)^{k-j}
\)

Regards,
Andrew Robbins

[Gottfried]

I believe I may have found a closed form for the third superroot / generalized LambertW function:
\(
{}^{3}W(v) = \log\left(\sqrt[3]{e^v}_s\right) = \sum_{k=0}^{\infty}
\frac{v^k}{k!} \sum_{j=0}^k
{k-1 \choose j}j(k-j)^{j-2}(-k)^{k-j}
\)

Regards,
Andrew Robbins

Hah, that sounds good, I'll try it tomorrow! (I've just seen formulae 96-100 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

[andydude]

I believe I found a slightly smaller or cleaner closed-form for the above function
\(
{}^{3}W(v) = \log\left(\sqrt[3]{e^v}_s\right) = \sum_{k=0}^{\infty}
\frac{v^k}{k!} \sum_{j=0}^{k-1}
{k-1 \choose j}(k-j-1)^j(-k)^{k-j-1}
\)

Regards,
Andrew Robbins

[andydude]

Did you see already whether it is possibly simply extensible to higher orders?

Gottfried

I tried doing something similar with superroot-4, but no luck, however, I found the coefficients by solving the equation

\( \exp_{\exp(u)}^{3}(z) = \exp(v) \)

so you should be able to add \( z^j \) to the above formula to solve this generalization. The above formula is just the special case when \( z=1 \).

[Gottfried]

[text updated]

Having not yet studied Andrew's formulae, I just played around with the idea of iterated superroots.

In this case, instead of \( x=H_2(y),x=H_3(y), ... \) with x being the second, third superroot of some y, I reconsidered simply the iterated second superroot - which is easier to implement, because \( x=H_2(H_2(y)) = \exp( LW(LW( \log(y)))) \) and for LW (the Lambert-W) there are easy implementations in M'tica and Pari/GP.

Of course, for \( x=H_2(H_2(y)) \) we have, that \( (x\^x)\^{(x\^x)}= x\^x\^(x+1) = y \) - where x is a bit smaller than \( H_3(y) \) (of course the latter is what I tried to approximate by some iteration).

Now I found the following amazing procedure. Consider for example \( y=3\^3\^3 \) .
Then compute \( x_1=H_2(y) ,x_2=H_2(x_1), x_3=H_2(x_2) , ... \) up to some limit.
Then the inhomogenuous exponentialtower / "nested exponentiation" (wikipedia) \( x_1\^ ... \^x_{n-2} \^x_{n-1} \^ x_n \approx y \) .

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 Lambert-W-implementation
   \\ gives the 2nd order superroot such that x=h2(z) then x^x = z                

y= 3^3^3            
  x = h2(y)
  vx=vectorv(32);
  for(k=1,#vx,  vx[k] = x;   x=h2(x))

print(Mat(vx )  )        \\ show entries of vx

   x=1
   forstep(k = #vx,1,-1,  x = vx[k]^x  )
print(x, " err= ", y-x  )                
\\   7625597484987  err= -7.16679172898 E-19

Nice...
Here the vectorv vx (read from left to right, then top down):

Code:

11.9551115478   2.59837825984   1.73428401000   1.45860478939
  1.32859289427   1.25423808991   1.20653347357   1.17350191467
  1.14935596533   1.13097611094   1.11653974291   1.10491388712
  1.09535875646   1.08737159082   1.08059919437   1.07478641134
  1.06974445983   1.06533075563   1.06143565139   1.05797348186
  1.05487637153   1.05208986123   1.04956976099   1.04727984662
  1.04519014815   1.04327566018   1.04151535801   1.03989143819
  1.03838872616   1.03699420970   1.03569666809   1.03448637486

Appendix: the code for Lambert-W, taken from wikipedia:

Code:

LW(x, prec=1E-80, maxiters=200) = local(w, we, w1e);        

w=0;        
for(i=1,maxiters,      
   we=w*exp(w);w1e=(w+1)*exp(w);          
   if(prec>abs((x-we)/w1e),return(w));
   w=w-(we-x)/(w1e-(w+2)*(we-x)/(2*w+2));                      
    );                
print("W doesn't converge fast enough for ",x,"( we=",we);              
return(0);

For y=27^27 we get the following vector vx of "stairs":

Code:

27.0000000000  3.00000000000  1.82545502292  1.49546396135
  1.34791839460  1.26596744805  1.21434913533  1.17905692100
  1.15349490568  1.13417277551  1.11907937973  1.10697800580
  1.09706811520  1.08880952478  1.08182500080  1.07584339381
  1.07066495590  1.06613938655  1.06215148458  1.05861150706
  1.05544853130  1.05260578496  1.05003729703  1.04770545446
  1.04557919154  1.04363262931  1.04184403978  1.04019504831
  1.03867001262  1.03725553455  1.03594007272  1.03471363248
  1.03356751591  1.03249411869  1.03148676386  1.03053956501
  1.02964731291  1.02880538111  1.02800964680  1.02725642431
  1.02654240873  1.02586462813  1.02522040273  1.02460730994
  1.02402315427  1.02346594142  1.02293385576  1.02242524082
  1.02193858225  1.02147249291  1.02102569984  1.02059703276
  1.02018541391  1.01978984917  1.01940942003  1.01904327661
  1.01869063135  1.01835075341  1.01802296368  1.01770663028
  1.01740116452  1.01710601728  1.01682067577  1.01654466054
...

where we see by the first two or three entries, that they are the consecutively the (iterated) second-order superroot of each other: y=27^27; 27=3^3; 3=1.82...^1.82... ; and so on and
y = 27^27 = 27 ^ 3 ^ 1.82... ^ 1.49... ^...

[tommy1729]

@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 base-change .... Well Maybe ...

Regards

Tommy1729

[andydude]

@Tommy1729
Thanks!

@Gottfried
By "inhomogenuous" do you mean "heterogeneous"?

Iterated superroots? I need some time to wrap my head around this...

[Gottfried]

Hi Andrew -

By "inhomogenuous" do you mean "heterogeneous"?

Well, this might also be correct. I simply mean that the power-tower 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).

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
\( z=y^y ; y=x^x; x=w^w; ... \)

Gottfried

[Gottfried]

The thing is solving (x_m ^ x_m)^[m] = y is only close to solving
X_n^^[n] = y ( n = m in value )

True. But having this way a (non-trivial) 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 re-combining them to a "nested exponential". We have not yet many examples of "nested exponentiations" with a meaningful outcome.
For instance, the construction of the Schroeder-function is based on (ideally) infinite iteration of the base-function to get a linearization. If we iterate the h2()-function infinitely, the curve of the consecutive values in an x/y-diagram (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 gamma-function, that was in one version putting together sequences of integer numbers weighting and repeating in a meaningful way; there is some infinite product-representation for his gamma-function I think I recall correctly... )

(see also the updates in my previous (introducing) posting)