Mixed Powers of Generating Functions

Lladser, Manuel

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006).

Résumé

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.

Détail

DOI : 10.46298/dmtcs.3501