Left associative exponentiation- an iteration exercise

[Gottfried]

A nice accidental result - although it is nearly trivial...

I tried, which matrix-operator would perform the left-associative exponentiation - well, this can easily be solved, but the appearance of this operator was much surprising to me.

First assume the factorially similarity-scaled matrices of Stirling numbers 2nd and 1st kind (as I described them many times). If S2 is that lower-triangle containing Stirling-numbers 2nd kind and S1 that containing 1st kind, then fS2F and fS1F are the factorially similarity scalings:
fS1F = dF^-1 * S1 * dF
fS2F = dF^-1 * S2 * dF

for instance:

Code:

´
     fS2F = fS1F^-1

     V(x)~ * fS1F = V(log(1+x))~

Now consider the matrix

Code:

´
    DPW_m = fS1F * dV(m) * fS2F

If we use it as matrix-operator we have the following transform of powerseries:

Code:

´
V(x-1)~        * fS1F  = V(log(x))~
V(log(x))~     * dV(m) = V(m*log(x))~ = V(log(x^m))~
V(log(x^m))~   * fS2F  = V(exp(log(x^m))-1) ~
                        = V(x^m -1)~

so

Code:

´
V(x - 1)~ * DPW_m  = V(x^m - 1)~

and integer iterations by integer powers of DPW_m

Code:

´
V(x - 1)~ * DPW_m^k = V(x^(m^k) - 1)~

If we interpret V(x-1) as a binomial-transform of V(x) then we can note this at the beginning and at end we have:

Code:

´
(V(x)~ * P^-1~)  * DPW_m^k   =  (V(x^(m^k))~ * P^-1 ~)

then

V(x)~ * (P^-1~  * DPW_m^k * P~) = V(x^(m^k))~

and denote the new product of matrices in the parenthese formally

Code:

´
   PW_m = P^-1 ~ * (DPW_m * P~)

then we have the operator for x-> x^m or, when k-times iterated, x -> x^(m^k), whenever the product P^-1~ * () is computable.

The much surprising aspect is the shape of DPW_m * P~: this is just the binomial-matrix where the columns represent the binomial-coefficients for the orders according to m - which also can be fractional.
For instance, m=2, DPW_2*P~ =

Code:

´
  1  1  1   1   1    1    1     1
  .  2  4   6   8   10   12    14
  .  1  6  15  28   45   66    91
  .  .  4  20  56  120  220   364
  .  .  1  15  70  210  495  1001
  .  .  .   6  56  252  792  2002
  .  .  .   1  28  210  924  3003
  .  .  .   .   8  120  792  3432

which gives, premultiplied by P^-1~ (here the multiplication is possible)

Code:

´
  1  .  .  .  .  .  .  .
  .  .  .  .  .  .  .  .
  .  1  .  .  .  .  .  .
  .  .  .  .  .  .  .  .
  .  .  1  .  .  .  .  .
  .  .  .  .  .  .  .  .
  .  .  .  1  .  .  .  .
  .  .  .  .  .  .  .  .

the "quasi"-diagonal matrix, which simply shifts all powers of x in V(x) to their 2*power-position V(x^2).
For instance, the even-indexed powers in [(1),x,(x^2),x^3,(x^4)...] get simply shifted to the consecutive positions [(1),(x^2),(x^4),(x^6),...], so we have a transform of x->x^2 and of all consecutive powers.



For fractional "bases of iteration" ( this means exponents here) m=1/2

Code:

´
  1         1  1        1  1        1  1        1
  .       1/2  1      3/2  2      5/2  3      7/2
  .      -1/8  .      3/8  1     15/8  3     35/8
  .      1/16  .    -1/16  .     5/16  1    35/16
  .    -5/128  .    3/128  .   -5/128  .   35/128
  .     7/256  .   -3/256  .    3/256  .   -7/256
  .  -21/1024  .   7/1024  .  -5/1024  .   7/1024
  .   33/2048  .  -9/2048  .   5/2048  .  -5/2048

the premultiplication by P^-1~ gives divergent results, cannot be done and must applied to the V(x)~-parameter instead.

The most interesting aspect is, that with integer m the column-vectors of PW_m are finite - so we can safely define the version with P^-1~ premultiplied - which gives, for integer m (and heights) - nicely the quasi-diagonal-matrices, which transform a vandermondevector V(x) into one of V(x^2), V(x^3) or the like...

And finally, the aspect, which really pleases me, is, that for fractional m we just get the correct column-vectors of fractional binomials - a smooth procedure, which I was looking for over the last monthes (unsatisfied with the hardcoded binomial-expressions otherwise)

Here another example with m=2/3, which means, each multiple of third power of x in the result is composed by smaller integer powers of x, and only the other powers are composed by the infinite sequences of according fractional binomially weighted powers of x:

Code:

´
  1          1          1  1         1          1  1          1
  .        2/3        4/3  2       8/3       10/3  4       14/3
  .       -1/9        2/9  1      20/9       35/9  6       77/9
  .       4/81      -4/81  .     40/81     140/81  4     616/81
  .     -7/243      5/243  .   -10/243     35/243  1    770/243
  .     14/729     -8/729  .     8/729    -14/729  .    308/729
  .   -91/6561    44/6561  .  -28/6561    35/6561  .  -154/6561
  .  208/19683  -88/19683  .  40/19683  -40/19683  .   88/19683

so, the spin-off-result here is:
the fractional m in fS1F*dV(m)*fS2F*P~ gives the correct composition for the matrix-operator, which performs the binomial-theorem for fractional exponents, and its (possibly fractional) powers give the expected formal powerseries for the expected binomial composition.