TITLE: Equation between the two branches of the lambert w function
QUESTION [5 upvotes]: My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r_1(y)$ and $r_2(y)$ of the equation $x^2=y$ fulfill the equation $r_1(y)=-r_2(y)$. So if one has computed one root, he already knows the second one by taking the negative of the computed root. It is also possible to calculate $W_0(y)$ by knowing $W_{-1}(y)$ and vice versa?
Note: I asked the question two years ago on math.stackexchange.com. Unfortunately I didn't get an answer or comment there. That's why I decided to reask the question here and I hope that's okay.
I read that questions shall not be migrated when they are older than 60 days. That's why I did not ask for migration on MSE...

REPLY [2 votes]: The two branches are related in a trascendental way. Their difference can be used to solve other trascendental equations.
If $y(x)=\frac{x}{1-e^x}$
then :
$x=W_0(ye^y)-W_{-1}(ye^y)=y-W_{-1}(ye^y)$ for  $-1< x <0 $
$x=W_{-1}(ye^y)-W_{0}(ye^y)=y-W_{0}(ye^y)$ for  $x < -1 $
See
http://www.apmaths.uwo.ca/~djeffrey/Offprints/SYNASC2014.pdf