TITLE: An identity for the Lambert $W$ function
QUESTION [5 upvotes]: Expressing the integral in  An integral identity in terms of residues, we come to the following supposed identity:
$$\sum_{k=-\infty}^\infty\frac1{1 + W_k(x)}=\frac12$$
for all $x\in(-1/e,0)$, where $W_k$ is the $k$th branch of the Lambert $W$ function.
How can this be proved?

REPLY [2 votes]: This identity is now established, since
An integral identity cited in the above post is now proved.