Is there any general formula that describes the sum and difference of Lambert W Function?

Question

I know $\ln(a)+\ln(b)=\ln(ab)$ and $\ln(a)-\ln(b)=\ln(\frac{a}{b})$ or $\ln(a^{b}) = b\ln(a)$ and other rules for the ln (Natural Logarithm). But are there any sum or difference laws for the lambert W Function (Product Log) i.e. any general formula for
$W(x) + W(y)$ or $W(x) - W(y)$ and so on.

Answer 1

We have

$$\begin{align} W(x) + W(y) &= W\Big(\big(W(x)+W(y)\big)e^{W(x)+W(y)}\Big)\\ &= W\Big(W(x)e^{W(x)+W(y)}+W(y)e^{W(x)+W(y)}\Big)\\ &= W\Big(W(x)e^{W(x)}e^{W(y)} + W(y)e^{W(y)}e^{W(x)}\Big)\\ &= W\Big(xe^{W(y)} + ye^{W(x)}\Big)\\ &= W\bigg(\frac{xy}{W(y)}+\frac{xy}{W(x)}\bigg)\\ &= W\bigg(xy\Big(\frac{1}{W(x)} + \frac{1}{W(y)}\Big)\bigg) \end{align}$$

Is this the kind of formula you're looking for?