Just some stuff I've found out about the equation...
Observation 1 : We write: ( where f appears k times )
Your equation then turns into: very suggestive, isn't it?
This clearly generalises to: now fix and set: solving the recurrence relation:
Setting: we have the relation: ( in particular your solutions correspond to the cases and the other )
Now, this implies: and going to the original equation we get the following restriction
So it seems it all comes down to solving that functional equation.
Observation 2: is injective. To see this, suppose then thus: but then and so
Observation 3: It follows from 2 that since we must have