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

1$\begingroup$ Thanks a lot. Do you have references for these formulas? $\endgroup$– joaopaAug 23, 2021 at 7:29

1$\begingroup$ Ira M. Gessel, Lagrange inversion, Journal of Combinatorial Theory, Series A 144 (2016), 212249, doi.org/10.1016/j.jcta.2016.06.018. $\endgroup$ Aug 23, 2021 at 14:16

$\begingroup$ I did not manage to use your formula to explicit the coefficients of the inverse of a power series (for the composition). Can you explain how to do that? Thanks in advance. $\endgroup$– joaopaAug 23, 2021 at 22:13

1$\begingroup$ To find the compositional inverse $f(x)$ of $g(x)$, let $R(t) = t/g(t)$ and solve $f=xR(f)$. $\endgroup$ Aug 23, 2021 at 22:19

$\begingroup$ Thanks a lot. Your answers are very useful. $\endgroup$– joaopaAug 23, 2021 at 23:01