In my research, I have come across a system of probability generating functions of the following form: $$H_1(x) = x A(H_1(x))B(H_2(x)) \text{,}$$ $$H_2(x) = x C(H_1(x))D(H_2(x)) \text{,}$$ and I am interested in the asymptotic power-series expansion of a third function, $F(H_1(x),H_2(x))$. For a system with only one equation, e.g. $h(x) = x \Phi(h(x))$, one has the Lagrange-Burmann formula: $[x^n]\{G(h(x))\} = \frac{1}{n}[h^{n-1}]\{G'(h)\Phi(h)^n\}$ (cf. Generatingfunctionology, page 167), which would apply if I didn't have both $H_1$ and $H_2$. Unfortunately, I don't see an obvious way to generalize the proof given in the book to the case that I have both $H_1$ and $H_2$. In the literature, I've found several references to a Multivariable Lagrange Inversion formula (e.g. 1987, Ira M. Gessel), which applies for a system of equations $$h_i(\vec{x}) = x_i g_i(\vec{h})\text{,}$$ however the form doens't quite apply, because each $h_i$ is multiplied by a distinct $x_i$, whereas my set of equations share the same $x$. Is there a generalization for a system of equations where the variable $x$ is shared between the generating functions? Is it possible to use the multivariable equation for the case of a single variable and multiple functions?
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$\begingroup$ See Theorem 4 of my paper. (It solves $h_i(\vec{x}) = g_i(\vec{h})$; the variable $x$ is part of $g_i$.) $\endgroup$– Ira GesselJul 9, 2019 at 13:24
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$\begingroup$ Thank you so much for the reply, this is exactly what I needed. $\endgroup$– Daniel KorchinskiJul 11, 2019 at 15:39
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