Abstract

An asymptotic estimate is given for the coe#cients of products of large powers of generating functions. This theorem and another local limit theorem which is useful for conditioning are applied to various combinatorial enumeration problems that involve multivariate Lagrange inversion. 1991 AMS Class. No. Primary: 41A63 Secondary: 05A16, 05C05, 41A60 ### ########### ###### ## ############# # ####### ### # 1. Introduction If f(0) #= 0 has a (possibly formal) power series expansion at 0, the equation w = xf(w) determines the power series w(x). Two forms of the Lagrange inversion formula are g n =[x n ] g(w)=[x n ] # g(x)f(x) n {(1 - xf # (x)/f (x)} # (1) =(1/n)[x n ] # xg # (x)f(x) n # , (2) where [x n ] h(x) denotes the coe#cient of the monomial x n in the power series h(x). We obtained asymptotics for g n from (2) for some types of formal power series [6]. When f has a nonzero radius of convergence, various authors have studied the asymptotics of [x n ] g...

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