Lagrange inversion for power-series with rational powers

Question

One can use Lagrange inversion to find the power series $F(x)$, which solves $F(x) = x(1+F(x)^p)$, where $p$ is a positive integer.

Now, what if $p$ is not an integer, but rather a positive rational number, say $p=7/3$?

As a concrete example, we are looking for a formal solution to $F(x) = x(1+F(x)^{7/3})$, but now, $F(x) \in \mathbb{C}[x^{1/3}]$.

The Lagrange inversion formula still seem to work in this case, that is, the function $$ F(x) := \sum_{r>0} x^r \left( [t^{r-1}]\frac{1}{r} (1+t^p)^r \right) $$ is a solution to $F(x) = x(1+F(x)^p)$, but now we must have $F(x) \in \mathbb{C}[x^{1/d}]$, where $d$ is the denominator of $p$, and the sum ranges over all positive integer multiples of $1/d$.

Is there some reference which proves this extension of Lagrange inversion?

Edit: I think I managed to prove that Lagrange inversion generalizes to this setting, i.e, instead of having $f,g \in \mathbb{C}[x]$, we have $f,g \in \mathbb{C}[x^{1/d}]$, and we wish to express the coefficients of $g$, in terms of the coefficients of $f$, where $f(g(x))=x$.

Answer 1

Let $p$ be any non-zero complex exponent. Let $f \in x\big(1+\mathbb C[[x]]\big)\subset \mathbb C[[x]]$ be the well-known compositional inverse of $x(1+x)^{-p}\in x\big(1+\mathbb C[[x]]\big)$ given by the Lagrange inversion: $$f(x)=\sum_{n=1}^\infty\frac1n {pn\choose n-1}x^n,$$

and consider its formal conjugate $F(x):=f(x^p)^{1/p}$, thus an element of $M:=x\big(1+\mathbb C[[x^p]]\big)$. Since $f$ solves $f(x)=x\big(1+f(x)\big)^p$ we have that $F$ solves in $M$ $$F(x)=x\big(1+F(x)^p\big).$$ This on the formal side; also, e.g. for small real nonnegative $x$, one gets a convergent series.

(Not needed here). There is also an easy extension of the Lagrange inversion formula that already works in $X^\alpha\mathbb C((X))$ as a $\mathbb C((X))$-module, for $\alpha\in\mathbb C$. I reported it in the wiki article. An application is e.g. the power series expansion of the complex powers of the Lambert function.

Answer 2

I managed to write up a proof based on these lecture notes (which cites Stanley's EC2)

The original proof uses residue calculus, and properties of analytic functions. This proof does not generalize to rational exponents, but the proof linked above, extends without any issues to "power series" with rational powers.