Half-iterate exp(z)-1: hypothese on growth of coefficients

[tommy1729]

ok here is my idea.

oh well or someone elses ofcourse there is alot of overlap.

anyway.

Lets say a taylor series with radius 0 but being analytic elsewhere is 

" analytic at some infinitesimals h "

As example f(x) = 2^^1 + 2^^2 x + 2^^3 x^2 + 2^^4 x^3 + ... 
is probably nowhere analytic.

But f(x)  = 2! + 3! x + 4! x^2 + 5! x^3 + ... is.

Now let that infinitesimal be a positive real infinitesimal.

that implies or assumes the taylor at 0 is an expansion at the positive axis.

now let tg(x) be the same as g(x) but the coefficients in absolute value.

( yes fake function ideas again )

so now we want to express tg(h) for positive real infinitesimal h...and have it as an upper bound.

To do so consider tg2(x) where tg2 is the same as g2 but the coefficients in absolute value.

g2(x) is equal to g(x) but expanded at 1 , so the taylor at 1 or g(x+1).

now

tg(h) coefficients a_n are given by the bounds 

tg(h) = tg2(x+1) with h = x , expanded at 0 by using binomium summing from tg2(x).

then 

g(x) coefficients are bounded by this tg(h) in absolute value.

that is the idea.

i know alot of assumptions and stuff.
but some logic.


regards

tommy1729