I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin for which $f(0)=0$ and $f'(0) \neq 0$ from the following Laplace transform method. All indeterminates and variables introduced below are considered real-valued.
Consider the polynomial $p(x) = \sum_{n=1}^N b_n \; \frac{x^n}{n!}$ with $N$ finite#. Impose the constraints $b_1>0$ and all other coefficients $b_n$ are such that $p(x)$ is strictly increasing for $x >0$, in particular $b_N > 0$, so $p(0) = 0$ and $p(\infty) = \infty$. Consequently, the compositional inverse $p^{(-1)}(x)$, defined globally by the curve $x=p(y)$, is a strictly increasing function for $x > 0$ with $p^{(-1)}(0)=0$ and $p^{(-1)}(\infty)=\infty$.
Now relabel $f(x)= p^{(-1)}(x)$ and $f^{(-1)}(x) = p(x)$. Then with $x\bar{t} = f^{(-1)}(t)$ and $x >0$,
(Eqn. $I$)
$$\int_0^{\infty} \; e^{-\frac{f(x \cdot \bar{t})}{x}} \; d\bar{t} = \int_0^{\infty}\frac{1}{x} e^{-\frac{t}{x}} \; df^{(-1)}(t)$$
$$= \int_0^{\infty}\frac{1}{x} e^{-\frac{t}{x}} \; (f^{(-1)})'(t) \; dt = \int_0^{\infty}\frac{1}{x^2} e^{-\frac{t}{x}} \; f^{(-1)}(t)dt .$$
(The last equality is evident from integration by parts or by inspection of the term by term Laplace transforms of the series expansion of the inverse function of the last two Laplace transforms.)
Expanding the middle equality of the equation in a series gives
(Eqn. $II$)
$$\int_0^{\infty}\frac{1}{x} e^{-\frac{t}{x}} \; df^{(-1)}(t) = \int_0^{\infty}\frac{1}{x} e^{-\frac{t}{x}} \; (f^{(-1)})'(t) \; dt$$
$$=\int_0^{\infty}\frac{1}{x} e^{-\frac{t}{x}} \;\sum_{n=0}^{N-1} b_{n+1} \frac{t^n}{n!} \; dt = \sum_{n=0}^{N-1} b_{n+1} \int_0^{\infty}\frac{1}{x} e^{-\frac{t}{x}} \; \frac{t^n}{n!} \; dt= \sum_{n=0}^{N-1} b_{n+1} \; x^n . $$
I've imposed $b_1 > 0$, so $\frac{f(\omega)}{\omega}= \sum_{n=1}^\infty a_n \frac{\omega^{n-1}}{n!} > 0$ for $\omega$ a sufficiently small positive real number, i.e., $a_1 > 0$, so
(Eqn. $III$)
$$ \int_0^{\infty} e^{-\frac{f(xt)}{x}} dt = \int_0^{\infty} e^{xt\partial_{\omega=0}} e^{-\frac{t \cdot f(\omega)}{\omega}} \; dt = \sum_{n=0}^{\infty} x^n \partial_{\omega=0}^n \int_0^{\infty} \frac{t^n}{n!} e^{-\frac{t \cdot f(\omega)}{\omega}} dt = \sum_{n=0}^{\infty} x^n \partial_{\omega=0}^n (\frac{\omega}{f(\omega)})^{n+1} .$$
Identifying the two results gives the basic Lagrange inversion formula
$$b_{n+1} = \partial_{\omega=0}^n \; (\frac{\omega}{f(\omega)})^{n+1}$$
$$ = \frac{1}{a_1^{n+1}} \partial_{\omega=0}^n \; (\frac{1}{1+\sum_{k\geq 1}c_k \omega^k})^{n+1} $$
$$= \frac{1}{a_1^{n+1}} \partial_{\omega=0}^n \sum_{k = 0}^{\infty} \; \binom{n+k}{k}(-R(\omega))^k= Prt_{n+1}(a_1,a_2,...,a_{n+1})$$
where $c_k = \frac{a_{k+1}}{a_1(k+1)!}$ and $R(\omega) = \sum_{k \geq 1} \; c_k \omega^k.$ The inversion partition polynomials $Prt_n$ are described in OEIS A134685.
Ultimately, using the Taylor series remainder (Lagrange form, see Taylor's Theorem) and reflections of curves through the horizontal axis of the $xy$-plane allow the Laplace transform method to be extended to find the Taylor series about the origin of the compositional inverse of any function $f$ analytic at the origin for which $f(0) = 0$ and $f'(0) \neq 0$.
(The LPT relation makes it particularly easy to derive the diff id $-\frac{\partial}{\partial b_{n+1}}\;p^{(-1)}(x)= \frac{\partial}{\partial x}\;\frac{(p^{(-1)}(x))^{n+2}}{(n+2)!}$ and other identities.)
