In view of Iosif Pinelis's question about $\sum_{k\in\Bbb Z}1/(1+W_k(x))$, I played a little with
$S(x,j)=\sum_{k\in\Bbb Z}1/W_k(x)^j$. It seems that if $x$ is a rational number with small denominator, so is $S(x,j)$. My quick and dirty implementation may be wrong, but I seem to find
for instance (values from $j=1$ to $j=5$):

For $x=-0.1$: $-19/2$, $80$, $-715$, $19180/3$, $-228675/4$

For $x=-0.2$: $-9/2$, $15$, $-115/2$, $665/3$, $-20525/24$

For $x=-0.3$: $-17/6$, $40/9$, $-235/27$, $1420/81$, $-34675/972$

Similarly for other values. Is this true, well-known, what is the proof ?