Asymptotic expansions for Taylor coefficients of the composition of two functions

We give an asymptotic expansion for the Taylor coefficients of L(P(z)) where L(z) is analytic in the open unit disc whose Taylor coefficients vary ‘smoothly’ and P(z) is a probability generating function. We show how this result applies to a variety of problems, amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal sequences.