base holomorphic tetration

[bo198214]

\( g(x)=\ln(a) (e^x -1 ) \), \( \tau(x)=\frac{x}{\ln(b)}+a \), \( \tau^{-1}(x)=\ln(b)(x - a) \), \( g = \tau^{-1}\circ f\circ \tau \).

\( g(x)=\ln(a)x + \frac{\ln(a)}{2} x^2 + \frac{\ln(a)}{6} x^3 + \dots \)

This gives the following unsimplified coefficients of \( g^{\circ t} \):
\(
{g^{\circ t}}_1 = \mbox{lna}^{t}\\
{g^{\circ t}}_2 = \left(\frac{1}{2} \, \frac{{({(\mbox{lna})}^{t}^{2} - \mbox{lna}^{t})}}{{(\mbox{lna}^{2} - \mbox{lna})}}\right) \mbox{lna}\\
{g^{\circ t}}_3 = \left(\frac{1}{2} \, \frac{{(\frac{{({(\mbox{lna})}^{t}^{2} - \mbox{lna}^{t})} \mbox{lna}^{t}}{{(\mbox{lna}^{2} - \mbox{lna})}} - \frac{{({(\mbox{lna})}^{t}^{2} - \mbox{lna}^{t})} \mbox{lna}}{{(\mbox{lna}^{2} - \mbox{lna})}})}}{{(\mbox{lna}^{3} - \mbox{lna})}}\right) \mbox{lna}^{2} + \left(\frac{1}{6} \, \frac{{({(\mbox{lna})}^{t}^{3} - \mbox{lna}^{t})}}{{(\mbox{lna}^{3} - \mbox{lna})}}\right) \mbox{lna}
\)

One can see that the coefficients are polynomials in \( \ln(a)^t \) with rational coefficients in \( \ln(a) \).
One needs to investigate whether \( f^{\circ t}(1)=\tau\circ g^{\circ t}\circ\tau^{-1}(1)=a+\frac{1}{\ln(b)}\sum_{n=1}^\infty {g^{\circ t}}_n (\ln(b)(1 - a))^n \) is analytic in \( b=e^{1/e} \) with \( a=\exp(-W(-\ln(b))) \).

I just wanted to unify the variables: with \( a = \ln(a)/\ln(b) \) we can write:
\( b[4]t = f^{\circ t}(1)=\tau\circ g^{\circ t}\circ\tau^{-1}(1)=\frac{1}{\ln(b)}\left(\ln(a)+\sum_{n=1}^\infty {g^{\circ t}}_n (\ln(b) - \ln(a))^n\right) \), \( \ln(b)\in (0,1/e) \), \( \ln(a) = - W(-\ln(b))\in (0,1) \) or shorter, setting \( x=\ln(b) \) and \( y=\ln(a) \)

\( e^x [4] t = \frac{1}{x}\left(y+\sum_{n=1}^\infty {g^{\circ t}}_n (x - y)^n\right) \), \( x\in (0,1/e) \), \( y=-W(-x)\in (0,1) \).

The thing is now that Lambert \( W \) has a singularity at \( -1/e \), i.e. if \( x=\ln(b) \) approaches \( 1/e \).

The question is whether this singularity gets compensated somehow by the infinite sum.

I want to further simplify the formula: with \( x = y e^{-y} = y/e^y \)
\( e^{ye^{-y}} [4] t = e^y\left(1+\sum_{n=1}^\infty {g^{\circ t}}_n (e^{-y} - 1)^n y^{n-1}\right) \), \( y\in (0,1) \)
where \( {g^{\circ t}}_n \) are polynomials in \( y^t \) with coefficients that are rational functions in \( y \).
I hope i dint put errors somewhere;