Does anyone have taylor series approximations for tetration and slog base e^(1/e)?

[bo198214]

I ask because I want to observe how logarithmic semi-operators behave for bases less than or equal to \( e^{1/e} \). I haven't a clue where I might going about getting these.

Hmm, interesting question.
The theory is that the regular fractional iterates of \( f(x)=\eta^x \) (\( \eta=e^{1/e} \)) have 0 convergence radius around the development (and fix)point e. From the theory of Écalle it follows that one can compute the regular Abel function \( \alpha \), i.e. the inverse of the superfunction \( \sigma(z)=f^{[z]}(z_0) \), as the following limit:

\( \alpha_\theta(z) = \lim_{n\to\infty} F_{\theta}(f^{[-\theta n]}(z)/e-1) + \theta n \)
where
\( F_\theta(z)=\frac{1}{3}\log(\theta z) - \frac{2}{z} \)
for \( \theta z > e \),
\( \theta=\pm 1 \).

(a value of \( \theta=-1 \) indicates the function which is real on the left side of e and \( \theta=+1 \) indicates the function which is real on the right side of e.)

Though this formula is usually designed to compute values of \( \alpha_\theta \) numerically, it can also be used to compute the powerseries of \( \alpha_\theta \), say at \( z=0 \).

To compute this we see that we have to calculate the formal powerseries of \( f^{[n]} \) (if we start with the case of \( \theta=-1 \)) and further then apply logarithm and reciprocal to this powerseries. The composition of two powerseries \( (g\circ h)(x)=g(h(x)) \) is possible with rational terms of the coefficients of \( g\circ h \) if h(0)=0.
But \( f(0)=\eta^0=1\neq 0 \). Despite we can construct \( f^{[n]} \) with only using compositions of powerseries \( H_n \) which satisfy \( H_n(0)=0 \).

This can be done with the following inductive definition:
\( H_1(x)=\eta^x - 1 \) and
\( H_{n+1}(x) = (\eta[4]n)^{H_n(x)}-1 \)

Verify that always:
\( f^{[n]}(x)=(\eta[4](n-1))*(H_n(x)+1) \)

for example:
\( f^{[2]}(x)]=\eta^{\eta^x} = \eta\eta^{\eta^x-1}=\eta*(\eta^{H_1(x)} - 1 + 1) = (\eta[4]1)(H_2(x)+1) \)

We see that the constant term of the powerseries of \( f^{[n]} \) is \( \eta[4](n-1) \) and so it converges to \( e \), hence the constant term of \( f^{[n]}(x)/e-1 \) is \( \epsilon=\frac{\eta[4](n-1)}{e}-1 \) and \( \epsilon\neq 0 \) (though \( \epsilon\to 0 \)). Hence we can apply the logarithm
powerseries which is
\( \log(x+\epsilon)=\log(\epsilon)+\log(1+x/\epsilon) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \epsilon^n} x^{n} \)
and also the reciprocal powerseries
\( \frac{1}{x+\epsilon} = \frac{1}{\epsilon} \sum_{n=0}^\infty \frac{(-1)^{n}}{\epsilon^n} x^n \).

Ok, when one applies all this stuff to the calculation of \( \alpha_\theta \) with \( \theta=-1 \), i.e. the Abel function which is real on \( x<e \), I get the following coefficients for n=1000 iterations:

Code:

2.03, 0.602, 0.248, 0.0942, 0.0351, 0.0130, 0.00482, 0.00178, 0.000657, 0.000242, 0.0000893, 0.0000329, 0.0000121, 4.47e-6, 1.64e-6, 6.05e-7, 2.23e-7, 8.20e-8, 3.02e-8, 1.11e-8

it seems however that they are accurate only up to 3 digits.
The corresponding super-function developed at 2.03 has then the coefficients:

Code:

0, 1.66, -1.14, 0.839, -0.656, 0.534, -0.448, 0.385, -0.337, 0.300, -0.270, 0.245, -0.225, 0.207, -0.192, 0.179, -0.168, 0.158, -0.149, 0.141