How to obtain an asymptotic expansion of a sequence from an analytic identity satisfied by its generating function

Abstract

AbstractLet fn be a sequence of nonnegative integers and let f(x): = Σn≥0 fn xn be its generating function. Assume f(x) has the following properties: it has radius of convergence r, 0 < r < 1, with its only singualarity on the circle of convergence at x = r and f(r) = s; y = f(x) satisfies an analytic identity F(x, y) = 0 near (r, s); for some k ≥ 2 F0.j = 0, 0 ≤ j < k, F0.k ≠ 0 where Fi is the value at (r, s) of the ith partial derivative with respect to x and the jth partial derivative with respect to y of F. These assumptions form the basis of what we call the typical and general cases. In both cases we show how to obtain an asymptotic expansion of fn. We apply our technique to produce several terms in the asymptotic expansion of combinatorial sequences for which previously only the first term was known.