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

TL;DR

This paper develops asymptotic estimates for coefficients in powers of a power series under specific conditions, enabling analysis of their positivity and asymptotic behavior for large indices.

Contribution

It introduces new asymptotic formulas for coefficients of $(f(z))^n$ when $k$ scales as $n^ heta$, extending previous results to a broader range of $k$ values.

Findings

Derived asymptotic estimates for coefficients in $(f(z))^n$ for $k o ext{large}$

Proved positivity of coefficients under certain conditions for large $n$ and $ heta$

Applied estimates to show non-negativity of certain exponential and trigonometric sums

Abstract

We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \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 $f\left( 0\right) \neq 0,$ $f^{\prime }\left( 0\right) =0.$ We also show that there is a positive number $\varepsilon <1$ (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 $\varepsilon <\delta <1$, 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.