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: $$\begin{array}{c|ccccc} & j = 1 & j = 2 & j = 3 & j = 4 & j = 5 \\ \hline x = -0.1 & -19/2 & 80 & -715 & 19180/3 & -228675/4 \\ x = -0.2 & -9/2 & 15 & -115/2 & 665/3 & -20525/24 \\ x = -0.3 & -17/6 & 40/9 & -235/27 & 1420/81 & -34675/972 \end{array}$$ Similarly for other values. Is this true? Is it well known? What is the proof?
Added: for $x=-\exp(-1)$ we seem to have $S(x,j)=(-1)^j2$ (I now believe that this is wrong, but close).
OK, I think I solved it, but I think it is interesting nonetheless: these are the coefficients of $T^{j-1}$ of $$1/2+(1/x)(1+T)/(\exp(-T)-T/x)\;.$$ This must be Lagrange inversion. When you set $T=-1$, you obtain Pinelis's identity $\sum_{k\in\Bbb Z}1/(1+W_k(x))=1/2$, although the inversion of summation is not justified.
Added again: the above (I must make clear that I have not proved it) is equivalent to the formal identity $$\sum_{k\in\Bbb Z}\dfrac{1}{W_k(x)-T}=\dfrac{1}{2}+\dfrac{1+T}{xe^{-T}-T}\;.$$ Interpreting $W_k(x)$ to be close to $2\pi ik$, one can play the same game as with the hyperbolic sine, and obtain for instance $$\prod_{k\in\Bbb Z}\left(1-\dfrac{T}{W_k(x)}\right)=e^{-T/2}-(T/x)e^{T/2}$$
In fact, doesn't this last identity give a proof of all the above, since by definition the roots of the RHS are the $W_k(x)$ ?
