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Superroots (formal powerseries)
#1
Recently I found Andrew's remark in "designing a tetration library", that the superrots were not yet well developed. Facts on superroots seem to be spread over various threads; so to have some collection under an expressive title I've put together some details, tending to compile more information from time to time as they appear.
(@admin: maybe that msg is better located in some related thread, for instance the tetration library thread)

Ok, I've put the text in plain text and am lazy to MimeTex it today, perhaps I'll rework it next days.


======================================================================
A short collection concerning superroots
======================================================================

Starting point is the nice powerseries for

Code:
´ g(x) = (1+x)^(1+x) - 1

Using the exponential-/logarithm-series for this we write first

Code:
´  g(x) = exp( log(1+x)*(1+x)) - 1

and get

Code:
´ g(x)   =   1*x + 1*x^2 + 1/2*x^3 + 1/3*x^4 + 1/12*x^5 + 3/40*x^6
           - 1/120*x^7 + 59/2520*x^8 - 71/5040*x^9 + 131/10080*x^10
           - 53/5040*x^11 + O(x^12)

This series has the nice property, that the constant term vanishes and also, that the linear term has coefficient 1, so g(0)=0 and g'(0)=1, so we can do some common operations with it: inversion, iteration, ... getting exact coefficients.
Now we define higher orders by something like chaining, which is not exactly iteration of g().

The sequence of functions

Code:
´    g(x,1) = (1+x)-1
     g(x,2) = (1+x)^(1+x) - 1
     g(x,3) = (1+x)^(1+x)^(1+x) - 1
     ...   = ...

gives similarly nice shaped powerseries, for instance

Code:
´    g(x,3) = (1+x)^(1+x)^(1+x)  -1
          =  1*x + 1*x^2 + 3/2*x^3 + 4/3*x^4 + 3/2*x^5 + 53/40*x^6
           + 233/180*x^7 + 5627/5040*x^8 + 2501/2520*x^9 + 8399/10080*x^10
           + 34871/50400*x^11 + O(x^12)

From this it is easy to define a sequence of functions for exponentialtowers of integer heights:

Code:
´  f(x,h) = g(x-1,h) + 1  = x^x^x^...^x    // h-occurences of x

Note, that this is in principle all well known and is merely a restatement of known results.


The unusual aspect with that sequence of powerseries is, that the leading coefficients stabilize when the height increases, and thus we have a "strange" behave when the height increases to infinity.
Example: we get the following table of coefficients (where the rows contain the coefficients for one height and each column is associated with one power of x):

Code:
´ 0    1    0    0       0     0       0          0             0          ...
  0    1    1    1/2    1/3    1/12   3/40      -1/120        59/2520      ...
  0    1    1    3/2    4/3    3/2   53/40     233/180      5627/5040      ...
  0    1    1    3/2    7/3    3    163/40    1861/360     33641/5040      ...
  0    1    1    3/2    7/3    4    243/40    3421/360     71861/5040      ...
  0    1    1    3/2    7/3    4    283/40    4321/360    102941/5040      ...
  0    1    1    3/2    7/3    4    283/40    4681/360    118061/5040      ...
  0    1    1    3/2    7/3    4    283/40    4681/360    123101/5040      ...
     ...

where the first column (containing zeros) represent the placeholders for the nonexistent constant terms. (The first row was appended to get a meaningfully interpretation for the "once"-iterate; it represents just g(x,1) = (1+x) -1 . The limit case for h->inf begins with the same coefficients as the last row of the table above)


------------------------------------------------------------------
Inversion

Since the g(x,h)-series have no constant term but a linear term with unit-coefficent, we can invert each of that g-series. Expressed by the appropriate f-function we get the superroot-powerseries for each integer height. Let's denote the inverse functions as gi() and fi(), then
for gi(x,2) we get

Code:
´ gi(x,2) = x - x^2 + 3/2*x^3 - 17/6*x^4 + 37/6*x^5 - 1759/120*x^6
            + 13279/360*x^7 - 97283/1008*x^8 + 654583/2520*x^9 - 10800299/15120*x^10
            + 75519317/37800*x^11 + O(x^12)

and a higher h, for instance

Code:
´ gi(x,12) = x - x^2 + 1/2*x^3 + 1/6*x^4 - 3/4*x^5 + 131/120*x^6
           - 9/8*x^7 + 1087/1260*x^8 - 271/720*x^9 - 2291/10080*x^10
            + 523/630*x^11  + O(x^12)

which again stabilizes for h->inf

Code:
Table of coefficients for gi(x,h), h=1..
------------------------------------------------------------------------------------
  0  1   0    0      0      0          0          0            0            0   ...
  0  1  -1  3/2  -17/6   37/6  -1759/120  13279/360  -97283/1008  654583/2520   ...
  0  1  -1  1/2    7/6  -17/4    821/120     -25/12  -56269/2520    52079/720   ...
  0  1  -1  1/2    1/6    1/4   -349/120     161/24    -2642/315       677/72   ...
  0  1  -1  1/2    1/6   -3/4    251/120      -45/8   13897/1260   -10891/720   ...
  0  1  -1  1/2    1/6   -3/4    131/120       -1/8   -5213/1260     8909/720   ...
  0  1  -1  1/2    1/6   -3/4    131/120       -9/8    2347/1260    -4231/720   ...
  0  1  -1  1/2    1/6   -3/4    131/120       -9/8    1087/1260      449/720   ...
  0  1  -1  1/2    1/6   -3/4    131/120       -9/8    1087/1260     -271/720   ...
  0  1  -1  1/2    1/6   -3/4    131/120       -9/8    1087/1260     -271/720   ...
  ....
-----------------------------------------------------------------------------------

======================================================================
The h'th superroot
======================================================================


The shown powerseries, formally seen, give the functons for the h'th superroots:

Code:
´ fi(x^x,2)   = x    = gi(x^x-1,2)+1
  fi(x^x^x,3) = x
  ...                      
  fi(x^^h,h)  = x

and the computation of the h'th superroot can be implemented by calls of the gi(x,h)-function:

Code:
´ ssrt(x,h) = fi(x,h) = gi(x-1,h) + 1

This gives, if convergent, the base b, which must be exponentiated h times to equal the given value x.

-----------------------------------------------------------------------
Convergence:

Concerning the range of convergence I don't have an idea yet. For instance for g(x,2) we can guess a rate of decrease similar to µ/k^2 where k is the index and µ some constant, so we should have a range of convergence for |x|<=1 only. For f(x,2) consequently we had then 0<x<=2 .
It seems, that the occuring divergences are not "too strong" so that we can extend the domain for x using Euler-summation to get meaningful approximations even if only 64 or 128 terms are known.

-----------------------------------------------------------------------
Interpolation to fractional orders:

The special form of the powerseries, where each k'th coefficient becomes constant when chaining-height h>=k this poses a new challenge for the interpolation to fractional heights.
I have currently no idea how to proceed here...

-----------------------------------------------------------------------

(should be continued)

Gottfried
Gottfried Helms, Kassel
Reply


Messages In This Thread
Superroots (formal powerseries) - by Gottfried - 10/25/2009, 12:09 PM
RE: Superroots (formal powerseries) - by andydude - 10/26/2009, 01:50 AM
RE: Superroots (formal powerseries) - by robo37 - 10/28/2009, 09:20 AM
Superroots article "WexZal" - by Gottfried - 10/29/2009, 05:56 AM
RE: Superroots article "WexZal" - by bo198214 - 10/29/2009, 12:08 PM
RE: Superroots article "WexZal" - by Gottfried - 10/29/2009, 01:55 PM
RE: Superroots article "WexZal" - by bo198214 - 10/29/2009, 10:10 PM
RE: Superroots (formal powerseries) - by Stan - 04/05/2011, 03:22 AM

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