Simplifying a recursive fraction involving Lambert function

Question

Is there a simplification for the following recursive fraction :

$$\frac{\frac{\frac{n}{W(n)}}{W\left(\frac{n}{W(n)}\right)}}{W\left(\frac{\frac{n}{W(n)}}{W\left(\frac{n}{W(n)}\right)}\right)}$$

The above formula uses a recursion 3 times. I'm looking for a simplification when we have such a finite recursion, for instance when this one appears $i$ times. I would like to remove the recursion, i.e. obtain a single fraction.

Thank you.

Answer 1

Since $u=W(u)e^{W(u)}$, it follows that $\frac u{W(u)}=e^{W(u)}$ and we get

$$\frac{\frac{\frac{n}{W(n)}}{W\left(\frac{n}{W(n)}\right)}}{W\left(\frac{\frac{n}{W(n)}}{W\left(\frac{n}{W(n)}\right)}\right)}=e^{W\left(e^{W(e^{W(n)})}\right)}$$

Which is the best simplification I can see.