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Mixed Powers of Generating Functions

Abstract : Given an integer $m \geq 1$, let $\| \cdot \|$ be a norm in $\mathbb{R}^{m+1}$ and let $\mathbb{S}_+^m$ denote the set of points $\mathbf{d}=(d_0,\ldots,d_m)$ in $\mathbb{R}^{m+1}$ with nonnegative coordinates and such that $\| \mathbf{d} \|=1$. Consider for each $1 \leq j \leq m$ a function $f_j(z)$ that is analytic in an open neighborhood of the point $z=0$ in the complex plane and with possibly negative Taylor coefficients. Given $\mathbf{n}=(n_0,\ldots,n_m)$ in $\mathbb{Z}^{m+1}$ with nonnegative coordinates, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of $z^{n_0}$ of the Taylor series of $\prod_{j=1}^m \{f_j(z)\}^{n_j}$, as $\| \mathbf{n} \| \to \infty$. The associated parameter-varying integral has a phase term with well specified properties that make the asymptotic analysis of the integral amenable to saddle-point methods: for many $\mathbf{d} \in \mathbb{S}_+^m$, these methods ensure uniform asymptotic expansions for $[z^{n_0}] \prod_{j=1}^m \{f_j(z)\}^{n_j}$ provided that $\mathbf{n}/ \| \mathbf{n} \|$ stays sufficiently close to $\mathbf{d}$ as $\| \mathbf{n} \| \to \infty$. Our method finds applications in studying the asymptotic behavior of the coefficients of a certain multivariable generating functions as well as in problems related to the Lagrange inversion formula for instance in the context random planar maps.
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Manuel Lladser. Mixed Powers of Generating Functions. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.171-182. ⟨hal-01184705⟩

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