# Generalization of Lagrange-Burmann to system of self-consistency equations

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?

• See Theorem 4 of my paper. (It solves $h_i(\vec{x}) = g_i(\vec{h})$; the variable $x$ is part of $g_i$.) – Ira Gessel Jul 9 at 13:24
• Thank you so much for the reply, this is exactly what I needed. – Daniel Korchinski Jul 11 at 15:39