Asymptotic estimate for the polynomial coefficients

TL;DR

This paper derives an asymptotic estimate for the coefficients of certain polynomial expansions and investigates their unimodality and maximum points relative to parameters, providing insights into their asymptotic behavior.

Contribution

It provides the first asymptotic estimate for polynomial coefficients binom{n,q}{cn} and explores their unimodality and maximum points based on experimental conjectures.

Findings

Asymptotic estimate for binom{n,q}{cn} as n  extgreater{} 0.

Conjecture on unimodality of binom{n,q}{cn} - binom{n,q-1}{cn}.

Identification of q values where the maximum occurs, related to  ext{log}_2 n.

Abstract

The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\binom {n,q}{cn}-\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor$ or $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\lfloor\log_2{n}\rfloor$ or $q=\lfloor\log_2{n}\rfloor+1$.