Re: Why is the tetration so much harder to extend than the factorial?

(prev. msg cancelled due to errors)

Am 04.03.2009 21:41 schrieb mike3:
On Mar 4, 8:06 am, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
....
Currently I'm focused to the idea of iteration-series.
Maybe this gives such a framework for the whereabouts of
functional iteration and fractional tetration and can include
the unfriendly bases too (instead of getting lost in improper
complex powerseries) At least this had a certain inherent
logic...


What's "iteration series", anyway?

what I earlier called "powertowerseries" or "tetra-series".
Remember, we had the binomial-method for interpolation to
fractional iterates. This binomial-method uses the
integer-height powertowers to formulate a composition of
terms to have fractional heights.

So (using (a:b) for binomial(a,b))

f°h(x) = sum{k=0..inf} ( (h:k) * sum {j=0,k} ((-1)^(k+j)*(k:j)*f°j(x) ))

according the given formulae. So we have in principle

f°h(x) = a0 * f°0(x) + a1*f°1(x) + a2*f°2(x) + ...

a series, which is constructed from integer-iterates similarly
as we have it in power-series with integer-powers of the x-parameter.
Moreover, this method is not only valid for powertowers (as far as
it converges at all) but for other functions as well, provided the
iteration to infinite height converges.

For tetration base e^(1/e) I use the matrix-logarithm for
fractional iterates - checking the properties of such
iteration-series I found, that the matrix-logarithm-approach
can be as well formulated in terms of such an iteration-series.
And is again not restricted to powertowers/tetration.

Recently I had an iteration-series (without assigning that
name to it) to approximate the log-function, such that

x + phi°(x) + phi°2(x) + phi°3(x) + ... = log(1 + x )
phi°(x) + phi°2(x) + phi°3(x) + ... = log(1 + phi(x))

(don't know the thread at the moment, I think it was in
december) and I considered the priciple, that this can
be expressed via

x = log(1+x) - log(1+phi(x))

log(1+phi(x)) = log(1+x) -x
phi(x) = exp(log(1+x)-x) -1
= e*(1+x)/e^(1+x) - 1

e*(1 +x)
phi(x) = --------- - 1
e^(1 +x)

Then the partial sums of the iteration-series of phi to
approximate the logarithm has a certain exotic flair...

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

If we have functions, which are expressed by a matrix-operator,
and their iterates by the powers of the matrix, and we have
a problem, which requires application of powerseries-rules
to the powers of the matrix-operators, then we can express
the same problem by an iteration-series of the according function,
where the powers of the matrices are converted in heights
of the iterated function.

So if I apply the mercator-series to the matrix-operator to get
its matrix-logarithm, then I can instead use the same coefficients
and apply them to the iterated functions and have again established
an iteration-series (using, for instance the entries of the
stirling-matrices for log(1+x) or for exp(x)-1)

It looks perfectly like the umbral-notation for calculus; the
differentation can be expressed by a simple-matrix-operator
(log of the pascalmatrix) and repeated differentation by
its powers, and the "mystic" of umbral-calculus is seemingly
simply founded by that matrix-power/function-iteration mechanics.

I'm fiddling to get it more specific this days.

The charme of the concept of "iteration-series" lies in
the concept, that only the values of the iterates of a
function are taken to construct, for instance, its
fractional iterates - have not yet the best expression
for that aspect, though.
I've just to begin to explore that area conceptually...

Gottfried
.

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