Functional equation for the lambert w function?

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I've been long with this question or trying to find something similar: Is there a functional equation of a reflection formula for the Lambert W function? The Lambert W function is the inverse function of $xe^x$.
I haven't had success with series and integrals representations of this function. I Try to use this expression $W(1+x)-W(1-x)$ but I did not get anywhere but seeing the graph does not seem like a very complicated function in the real plane at least. Thanks for any progress