Estimating the kth coefficient of (f (z))n when k is not too large

We derive asymptotic estimates for the coefficient of zk in (f (z))n when n→∞and k is of order nδ, where 0 < δ <1, and f (z) is a power series satisfying suitable positivity conditions and with f (0) = 0, f (0) = 0. We also show that there is a positive number ε < 1 (easily computed from the pattern of nonzero coefficients of f (z)) such that the same coefficient is positive for large n and ε <δ<1, and admits an asymptotic expansion in inverse powers of k. We use the asymptotic estimates to prove that certain finite sums of exponential and trigonometric functions are non-negative, and illustrate the results with examples. © 2007 Elsevier Inc. All rights reserved