Some results on the asymptotic behaviour of coefficients of large powers of functions

Abstract

We review existing results on the asymptotic approximation of the coefficient of order n of a function $\text{ƒ(Z)}{}^{\mathrm{d}}$, when n and d grow large while staying roughly proportional Then we present extensions of these results to allow more general relationships between n and d and to take into account a multiplicative factor ψ(z), that may itself include ‘large’ powers of simpler functions. A common feature of all the results of the paper is the use of a saddle point approximation; in particular we show that an approximate saddle point can give simpler results, and we characterize precisely how far from the exact value this approximate saddle point can be.