Asymptotic estimate for the polynomial coefficients

Abstract

The polynomial coefficient $\left(\genfrac{}{}{0ex}{}{n,q}{k}\right)$ 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 $\left(\genfrac{}{}{0ex}{}{n,q}{cn}\right)$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\left(\genfrac{}{}{0ex}{}{n,q}{cn}\right)-\left(\genfrac{}{}{0ex}{}{n,q-1}{cn}\right)$ 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 $q=\lfloor {\log }_{2}n\rfloor +1$.

arXiv comment9 pages