How do I cite this document and does it say what I think it says?

[Chenjesu]

I appreciate the work but the question was not limited to fixed points since the W function has a more general relationship to tetration. I looked on wikipedia and noticed that for some reason the Taylor series for the -1 branch is drastically more complicated, and so I was wondering if it has a simpler series representation.

The nice thing about this particular implementation of LambertW for the W-1 and W0 branch pair, is that it has very nice convergence properties.  For example, this LambertW series converges for all z where 0.0016<abs(z)<84, plus many other points points with abs(z)<197.  Normally, this series would be used as a seed along with Newton's method.  The authors also give a closed form for the coefficients of the series in their paper (see below).

\( W(z)=-\text{wseries}\left(\pm\sqrt{-2(\ln(-z)+1)}\right) \)
\( a_0=1;\,a_1=1; \)
\( a_n=\frac{1}{n+1}\,\left(a_{n-1}-\sum_{k=2}^{n-1} k\,a_k\,a_{n+1-k}\right) \)
\( \text{wseries}=\sum_{n=0}^{\infty}a_n\,x^n;\;\;\;\text{wseries}=\text{xfixed}+1+\frac{x^2}{2};\;\;\; \) relationship to my xfixed series

Anyway, -0.00069 at the limit of convergence is not zero, though it corresponds to an upper fixed point of ~13817 for b=1.00069.  So the question is how does the Lambert -1 branch singularity behave near z=0, and is there an asymptotic?

So I don't understand exactly what this wseries and xpoint is supposed to mean, I've never seen anyone use that notation. If it exists then what is relevant is a typical Taylor series (or some kind of functional series) just with a standard sum from n_0 to infinity for the -1 branch and that's it, nothing fancy, no fixed points or computational approximations.

[sheldonison]

So I don't understand exactly what this wseries and xpoint is supposed to mean, I've never seen anyone use that notation. If it exists then what is relevant is a typical Taylor series (or some kind of functional series) just with a standard sum from n_0 to infinity for the -1 branch and that's it, nothing fancy, no fixed points or computational approximations.

wseries is just the Taylor series from this paper[45] ; we can express LambertW as follows where for negative z the +sqrt is the (-1) branch the Op asked about.
\( W(z)=-\text{wseries}\left(\pm\sqrt{-2(\ln(-z)+1)}\right) \)
\( W=-\text{wseries}\circ f;\;\;\;f=\pm\sqrt{-2(\ln(-z)+1)};\;\;\; \)  I'm not sure what non-standard notation I'm using; this is just function composition ...

Post#28 has the first 16 terms of the wseries Taylor series.  The recursive formula for the coefficients was provided in post#30.  wseries has the same a_3 to a_oo Taylor series coefficients as my xfixed series from post#27 which is before I discovered Corless's paper.  

My best guesss is that perhaps Chenjesu just views this as too complicated a solution, and he is looking for a simpler series.  The simpler LambertW Taylor series for the main branch at z=0 only has a radius of convergence of 1/e, and won't work for the fixed points of any base>exp(1/e).  So if that is Chenjesu's complaint, then yes, this is a more complicated series, but it is much more powerful since it gives the both the main branch and the (-1) branch, and since it converges for a fairly large subset of the complex plane.  Since the (-1) branch is only real valued at the real axis from -1/e to 0, and it has a really complicated singularity at 0, so there is no hope of getting any series in x centered at x=0.  Since the (-1) branch also has a square root branch at -1/e, that requires a square root term in the composition, so that can't be a simple series either.   There probably aren't any other rational x,W(x) pairings besides at -1/e.  The approach from Corless's paper has rational coefficients, and a square root in the substitution and is re-centered so that z=-1/e is mapped to zero; that might be the best that we can do for the (-1) branch.