Solve using some knowledge of "involuntary functions." For example, if for nonzero then again. Similarly, if then again. This last one is what we want. So
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