x exp(x) musing

[BenStandeven]

let f( x exp(x) ) * exp(x) * (x+1) = f(x)

then F( x exp(x) ) = F(x) + C

inversing ->

G(x+C) = G(x) * exp(G(x))

now notice G(x + n C) =
G(x) * exp(G(x)) * exp(G(x))^exp(G(x)) * ... n times.

thus exp(G(x)) ^^ n = G(x + n C) / G(x + n C - C) ( Form 1 )

Actually, I think you get exp(G(x)) ^^ 2^n = exp(G(x + n C)) = G(x + n C)/G(x + n C - C), where the operation is "symmetric" tetration (referring to the association method). I don't see any obvious way to get the usual "top down" tetration from this.

and maybe f(x) can be found by current attempts for tetration.

and G(x) might be found by inversing the ( modified ) carleman matrix of F(x)

It would probably be simpler to find G directly.

notice also that x exp(x) has a unique real fixed point !!

Moreover, it is at zero, so it doesn't even need shifting.