Estimating the $k$th coefficient of $(f(z){)}^n$ when $k$ is not too large

Abstract

We derive asymptotic estimates for the coefficient of $z^k$ in ${\left(f\left(z\right)\right)}^n$ when $n\to \infty$ and $k$ is of order $n^{\delta }$, where $0<\delta <1,$ and $f\left(z\right)$ is a power series satisfying suitable positivity conditions and with $f\left(0\right)\ne 0,$ $f^{\prime }\left(0\right)=0.$ We also show that there is a positive number $\epsilon <1$ (easily computed from the pattern of non-zero coefficients of $f\left(z\right)$) such that the same coefficient is positive for large $n$ and $\epsilon <\delta <1$, 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.

18 pages