Mixed powers of generating functions
Abstract
Given an integer m>=1, let || || be a norm in Rm+1 and let S denote the set of points with nonnegative coordinates in the unit sphere with respect to this norm. 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 a vector n=(n0,...,nm) with nonnegative integer coefficients, 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 (f1(z))n1...(fm(z))nm, as ||n|| tends to infinity. 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 directions d in S, these methods ensure uniform asymptotic expansions for the Taylor coefficient of zn0 of (f1(z))n1...(fm(z))nm, provided that n/||n|| stays sufficiently close to d as ||n|| blows up to infinity. 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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