Iteration series

[Gottfried]

Hi -

as I mentioned in the earlier thread

http://math.eretrandre.org/tetrationforu...hp?tid=186

the idea of series of powertowers, which define fractional heights, excited me much. After the translation of the Binomial-formula into my matrix-concept there was no progress in this; but after some fiddling last days, it seems, that I found the key to translate and thus to apply at least the core concepts in the "matrix-method" also to such powertower-series.
I'll call them from now more generally "iteration-series" because they also occured in some iteration-exercises with iterable functions other than tetration.
I defined in my matrix-concept the vandermonde-vector V(x) as infinite vector of powers of x and got this way a very handy way to formulate functions of powerseries, compositions and iterations, as the forum-fellows know already (at least my opinion in this regard :cool.
While

\( V(x) = colvector_{r=0}^{\infty} \lbrace x^r \rbrace \)

and have called this "Vandermonde-vector" I define now

\( IT(x) = colvector_{h=0}^{\infty} \lbrace f^{\circ h}(x) \rbrace \)

as "iteration-vector" where the related function f(x) shall be defined in the near context.

If it is needed to start at a different h_0 then I extend the definition to


\( IT(x,h_0) = colvector_{h=0}^{\infty} \lbrace f^{\circ h_0 *h}(x) \rbrace \)

so a change of h_0 to, say 1/2 changes the stepwidth to 1/2 (and the second element of the vector has the value of f°(1/2)(x))


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

I cannot yet post the full procedere of my computations of last days; but the first main result is that, similarly to the nice binomial-formula (Newton/Woon/Henryk...:-) ) I can show that the computation of fractional heigths can be done based on IT-vectors/-series the same way as fractional powers using logarithms, let me call it here: the function-logarithm- or stirling-formula.

Instead of fractional binomials, as in the binomial-formula, I can use the Stirling-numbers (which define the logarithmic- and exponential-series) on IT-series/-vectors, as one would do this with powerseries/Vandermonde-vectors.

I do not yet recognize, whether this gives an improvement of convergence compared to the binomial-formula for fractional iterates, I'll see tomorrow.

But unfortunately, it lacks the same problem as the binomial-formula, that if we want to use an IT-(iteration)-series, then the divergence-problem, if it exists, gets us very early, and with little hope to have a faster approximation-/summation-method.

It's late, I'll continue this tomorrow.

Gottfried

[Gottfried]

I'll explain the use of iteration-series in more detail.

The formula for the binomial-method, (which Henryk announced to the forum last year, was shown by Woon, and earlier by Comtet who credits again earlier use by Jabotinsky) is
\( \hspace{24}
f^{\circ h}(x) = \sum_{k=0}^{\infty}\left( (^h_k) \left( \sum_{j=0}^k (-1)^{k-j} (^k_j) f^{\circ j}(x) \right) \right)
\)

The most straigthforward translation of my application of the matrix-method into the similar serial notation is perhaps the following. I'm writing \( \hspace{24} [x]_0^m \) for the pochhammer-symbol x(x-1)(x-2)...(x-m)

derived from the logarithmic-formula exp(h*log(...)) using the matrices of Stirling-numbers I got
\( \hspace{24}
f^{\circ h}(x) = 1 + \sum_{k=0}^{\infty} \left(
\left(\sum_{j=0}^k (-1)^{k-j} (^k_j) f^{\circ j}(x) \right)
* [h]_0^{k-1} /k!
\right)
\)

[update] Upps... second read: seems I just reduced the combined exp( h* log() ) formulae to arrive at the same as above But well, I'll leave it as it is, the main focus is the following) [/update]

---

Well, it's not my intention at the moment to compare approximation-rates and convergence issues. It is just to introduce the translation from the concept in my matrix-formula (which is taylored for algebra with formal powerseries) to the equivalent formulation for iteration-series, where only integer-iteration-heights are required to arrive at (and possibly further use) fractional heights.

The whole concept is then much more easier to read (and write) if we introduce an "umbral"-like notation, adapted to iteration-series.

So my proposal tonight is,to write \( \left[f^{\circ}(x) + I^{\circ}\right]^m \) meaning the binomial-formula, in which after expansion the powers are understood as heights. (I is here the identity-operator.)

So \( \hspace{24} [f^{\circ}(x) +I^{\circ}]^3 \) expands formally to

\( \hspace{24}
[f^{\circ}(x) +I^{\circ}]^3 = f^{\circ 3}(x)I^{\circ 0} +3 f^{\circ 2}(x)I^{\circ 1}
+ 3 f^{\circ 1}(x)I^{\circ 2} + f^{\circ 0}(x)I^{\circ 3} \\
\hspace{48}
= f^{\circ 0}(x) +3 f^{\circ 1}(x) + 3^{\circ 2}(x) + f^{\circ 3}(x)
\)

and the analogue interpretation may be understood by the notation \( \log\left[f^{\circ}(x)\right] \),\( \exp\left[f^{\circ}(x)\right] \) for log(), exp() and other functions, which are expressed by powerseries.

Let us also omit the "(x)" at the function, to make things shorter.
Then with this newly introduced Iteration-Umbral we express the above binomial-formula

\( \hspace{24}
f^{\circ h} = [I^{\circ}+ (f^{\circ}-I^{\circ})^{\circ}]^h \\
\hspace{48} \left( = \sum_{k=0}^{\infty}\left( (^h_k) (f^{\circ}-I^{\circ})^k \right) \right) \\
\hspace{48} \left( = \sum_{k=0}^{\infty}\left( (^h_k) \left( \sum_{j=0}^k (-1)^{k-j} (^k_j) f^{\circ j} \right) \right) \right)
\)

and the logarithmic formula

\( \hspace{24}
f^{\circ h} = \exp\left[h * \log\left[I^{\circ}+ (f^{\circ}-I^{\circ})^{\circ}\right] \right]
\)

May be, the short-notation can further be polished, for instance first I thought to use an additional placeholder at the formal tiny iteration-circle - maybe such expliciteness is sometimes required in more complicated places.

Gottfried

[bo198214]

The formula for the binomial-method, (which Henryk announced to the forum last year, was shown by Woon, and earlier by Comtet who credits again earlier use by Jabotinsky) is

Woon used linear operators he doesnt talk about functions.
Jabotinsky (and also Comtet I guess) derived a similar (finite) formula for power series with \( f_0=0 \), \( f_1=1 \).

The whole concept is then much more easier to read (and write) if we introduce an "umbral"-like notation, adapted to iteration-series.

Yes, its kinda umbral because generally \( (f+\text{id})^{\circ n}\neq \sum_{m=0}^n \left(n\\m\right) f^{\circ n} \).

[Gottfried]

Hi,

I'm looking at iteration-series again, and with the intention to apply the concept of "fractional summation bounds" (as Markus Müller has described this here) meaningfully to that series - perhaps to get another notion of fractional iterates.
But also in general I'm much interested, what properties such series may have.
Here is a step into it. I have it in winword so far (this can at least convert it to html), so you can see it in the browser:
Iteration series 0908
There is a certain aspect in that series which seems to be connected with the slog - but currently I'm a bit stuck and my creativity has a rest today. Perhaps someone here can give a further leading impulse...

Gottfried