Multivariate Asymptotics for Products of Large Powers with Applications to Lagrange Inversion

An asymptotic estimate is given for the coefficients 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 the electronic journal of combinatorics 6 (1999), #R8 2 1. Introduction If f(0) 6= 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 gn = [x n ] g(w) = [x n ] i g(x)f(x) n f(1 \Gamma xf 0 (x)=f(x)g j (1) = (1=n)[x n ] \Gamma xg 0 (x)f(x) n \Delta ; (2) where [x n ] h(x) denotes the coefficient of the monomial x n in the power series h(x). We obtained asymptotics for gn from (2) for some types of formal power series [6]. When f has a nonzero radius of convergence, various authors have studied the asymp..