Does there exist an analog of Lagrange inversion formula in positive characteristic? Obviously, the formula is still valid for coefficient with index not divisible by the characteristic, but for the other ones I did not manage to find one.



Lagrange inversion formula in positive characterisic
There are many forms of Lagrange inversion. The ones that don't involve division by integers are valid in positive characteristic. For example:
Given a power series $R(t)$, there is a unique power series $f=f(x)$ such that
$f(x) = x R(f(x))$, and for any Laurent series $\phi(t)$ and $\psi(t)$ and any integer $n$ we have


$$[x^n]\phi(f)=[t^n]\bigl(1-tR'(t)/R(t)\bigr)\phi(t)R(t)^n$$
and
$$[x^n]\frac{\psi(f)}{ 1-xR'(f)}=[t^n]\psi(t)R(t)^n.$$


