I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with the implicit equation $$u(z) = tw(u(z))$$ where $w$ is a specified generating function for a formal power series. The goal is to solve for the generating function $u$ in the formal parameter $z$. When $w$ is sufficiently analytic and convergent near the origin, the Lagrange Inversion formula states that $$[z^n]{u} = \frac{1}{n!}( \frac{d^{n-1}}{du^{n-1}} w^n )$$ I am interested in obtaining an analogous (i.e. direct) formula for the same coefficients in the case where we start with $$u(z) = zw(u(z^\gamma))$$ with $\gamma \in \mathbb{R}$. By computing the first few terms, I find that, unlike in the original case the coefficients $[z^n]{u}$ can not be expressed as polynomials in derivatives $u$, but rather are rational polynomials in in derivatives $u$ (and also in gamma). This suggests to me that it may be possible to obtain an analog of the inversion formula in terms of some appropriate rational function of u(z) and u(\gamma z). However, the proof of the Lagrange inversion formula I am aware of explicitly uses the fact that $w$ is polynomial, which has impeded my efforts to direct apply the analogy. Is it possible to generalize to my case of interest?

P.S. The context in which I obtain this problem may be of interest. The "skewed" formula arises in the construction of approximate mean field solutions to $x\in \mathbb{R}^N$ in $(I+\gamma A)x=b$ where $A$ is the adjacency matrix of a Watt-Strogatz-Newman type directed random network. I obtain an expression for the moment generating function $X(t)$ of the random variable $x_i$ as $X(t) = e^tA(X(-\gamma t))$ where $A(z)$ is the factorial moment generating function of the in-degree of the network. In the most basic case $A(z) = (1-p+pz)^{N-1}$. The role of $\gamma$ here is the weight and of the randomness in the linear system of equations.

I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with the implicit equation $$u(z) = tw(u(z))$$ where $w$ is a specified generating function for a formal power series. The goal is to solve for the generating function $u$ in the formal parameter $z$. When $w$ is sufficiently analytic and convergent near the origin, the Lagrange Inversion formula states that $$[z^n]{u} = \frac{1}{n!}( \frac{d^{n-1}}{du^{n-1}} w^n )$$ I am interested in obtaining an analogous (i.e. direct) formula for the same coefficients in the case where we start with $$u(z) = zw(u(z^\gamma))$$ with $\gamma \in \mathbb{R}$. By computing the first few terms, I find that, unlike in the original case, the coefficients $[z^n]{u}$ can not be expressed as polynomials in $w,w',w'',\ldots, w^{(n)}$, but rather are rational polynomials in $w,w',w'',\ldots, w^{(n)}$ (and also in $\gamma$). This suggests to me that it may be possible to obtain an analog of the inversion formula in terms of some appropriate rational polynomial in $w,w',w'',\ldots, w^{(n)}$. However, the proof of the Lagrange inversion formula I am aware of explicitly uses the fact that $w$ is polynomial, which has impeded my efforts to directly apply the analogy. Is it possible to generalize to my case of interest?

P.S. The context in which I obtain this problem may be of interest. The "skewed" formula arises in the construction of approximate mean field solutions to $x\in \mathbb{R}^N$ in $(I+\gamma A)x=b$ where $A$ is the adjacency matrix of a Watt-Strogatz-Newman type directed random network. I obtain an expression for the moment generating function $X(t)$ of the random variable $x_i$ as $$X(t) = e^tA(X(-\gamma t))$$ where $A(y) = \sum_a' P(a')y^{a'}$ is the factorial moment generating function of the in-degree of the network. The moment generating function here is related to the ordinary generating functions in the statement of Langrange's inversion formula through the $z=e^t$ (it is both conventional and convenient to transform the variables in this way). As such, the mean value of $x_i$ is given $X'(0)$, the variance is given by $X''(0)-(X'(0))^2$ and so on. For example, in the case where the in-degree at every node is some constant $a$ with probability one, we have $A(y) = y^a$. One finds that $$ X'(0) = \frac{A(X(0))}{1+\gamma A'(X(0))} ~~~~~~= \frac{1}{1+\gamma a} $$ which could have been deduced from the original matrix equation by symmetry considerations. Continuing in the same fashion, one finds that $X''(0) = (X'(0))^2$, so that the variance is zero, which should have also been expected, since the equation in this case determined by the value of $a$. However, in the most common case of interest, we will allow every edge in the network to be realized independently with probability $p$. In this case $A(y) = (1-p+py)^{N-1}$.