closed form for regular superfunction expressed as a periodic function

[bo198214]

it only has a formal powerseries that converges at the integers but has radius 0 hence not analytic.

Ya but it is analytic outside the fixed point. There is one half-iterate to the right of the fixed point and one to the left, which are different functions, i.e. no analytic continuations of each other. Both iterates have the asymptotic power series expansion at the fixed point that one gets when calculating the *formal* powerseries (which has 0 convergence radius).