The Lambert's Omega function has 2 real branches denoted by $W_{-1}(x)$ and $W_0(x)$ and it represents the solution(s) of the equation $xe^x=a$.
I learned that this function can be generalized and for every $n\in \Bbb R$ so that $W_n(x)$ in the solutions to the equation $xe^x+nx=a$.
My problem is this: if I set $n=0$ I get the original equation so everything makes sense to me but if I set $n=-1$ do I get again my original definition of the branch $W_{-1}(x)$ ?
My equation becomes this: $xe^x-x=a$
Is it solvable with the "classical" Lambert's function (the negative branch of it) ? Probably I'm missing something very trivial but I can't see how to come back to the original definition of $W(x)$ this time.
This is a graphical answer to the question :
I don't know if there is a standardized symbol for this sort of generalized Lambert-W function. The symbols used above might be non-standard.