# Which values of Lambert W function are rational?

I mean the branch of Lambert-W function when it is single-valued (The $W_0$ function): Defined on $x\geq \dfrac{1}{e}$ and by the relation $w = W(x) \iff x = we^{w}$.

I know that $0$ is one of the rational values, since $W(0)=0$, because $0e^{0} = 0$. I wonder what are the others?

I see here (https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf) that I can write $W_0(x) = \displaystyle\sum_{n=1}^{\infty} \frac{(-n)^{n-1}}{n!} x^n$, then it is analytic around $0$, and the range is $[-1,\infty)$, so by the Intermediate Value Theorem there must be infinite values where $W(x)$ is rational, but I have no idea how to analyse that.

Any help would be appreciated.

In addition to $W(-e^{-1}) = -1$ and $W(0)=0$, there are infinitely many: if $p$ and $q$ are integers, then $x = (\frac pq) \exp(\frac pq) \implies W(x) = \frac pq$. This follows from the definition of $W(x)$, which is $$W(x) \cdot e^{W(x)} = x.$$ Substituting $x = (\frac pq) \exp(\frac pq)$, $$W(x) e^{W(x)} = \left(\frac pq\right) \exp\left(\frac pq\right). \tag{1}$$ Comparing both sides of $(1)$, clearly $$W(x)=\frac pq.$$