On The Taylor Coefficients Of The Composition Of Two Analytic Functions

. We give an asymptotic formula for the Taylor coefficients f n of f(z) = l(h(z)) where l(z) is analytic in the unit disc whose Taylor coefficients l n vary `smoothly' and h(z) is analytic in a larger disc. We show that under mild conditions on h(z) , f n ¸ oel [oen] as n ! 1 where oe = 1=h 0 (1) . Applications to renewal theory are also discussed. 1. Introduction Asymptotic enumeration usually involves estimating coefficients of a generating function f(z) which satisfies some functional equation. In many cases such a functional equation can be reduced to the form f(z) = l \Gamma h(z) \Delta ; where the function l(z) is frequently known and its Taylor coefficients l n are nonnegative and usually satisfy a certain regularity condition (see Bender [1], Meir and Moon [11]). In this paper we make the following regularity condition on l n which occurs in many applications: (y) l n+[ p n ] ¸ l n as n !1 for all 2 R: For instance, the cases l n = 1 and l n = 1=n from renewal ..