Mixed powers of generating functions

Given an integer m ≥ 1, let ‖·‖ be a norm in R and let S+ denote the set of points d = (d0, . . . , dm) in R with nonnegative coordinates and such that ‖d‖ = 1. Consider for each 1 ≤ j ≤ m a function fj(z) that is analytic in an open neighborhood of the point z = 0 in the complex plane and with possibly negative Taylor coefficients. Given n = (n0, . . . , nm) in Z with nonnegative coordinates, we develop a method to systematically associate a parametervarying integral to study the asymptotic behavior of the coefficient of z0 of the Taylor series of Qm j=1{fj(z)} nj , as ‖n‖ → ∞. 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 d ∈ S+ , these methods ensure uniform asymptotic expansions for [z0 ] Qm j=1{fj(z)} nj provided that n/‖n‖ stays sufficiently close to d as ‖n‖ → ∞. 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.