Given an integer m≥1, let | · | be a norm in ℝm+1 and let 𝕊+m denote the set of points d=(d0,…,dm) in ℝm+1 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 𝕊m+1 with nonnegative coordinates, we develop a method to systematically associate a parameter-varying integral to study the asymptotic behavior of the coefficient of zn0 of the Taylor series of ∏j=1m {fj(z)}nj, as |&bfn;|→∞. 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+m, these methods ensure uniform asymptotic expansions for [zn0]∏j=1m {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.