**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...