Confirming Two Conjectures of Su and Wang

Abstract

Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For $n\ge k\ge 0$ and $b>a>0$, we show that the finite sequence $C_j=\left(\genfrac{}{}{0ex}{}{n+ja}{k+jb}\right)$ is a Pólya frequency sequence. For $n\ge k\ge 0$ and $a>b>0$, we show that there exists an integer $m\ge 0$ such that the infinite sequence $\left(\genfrac{}{}{0ex}{}{n+ja}{k+jb}\right),j=0,1,...$, is log-concave for $0\le j\le m$ and log-convex for $j\ge m$. The proof of the first result exploits the connection between total positivity and planar networks, while that of the second uses a variation-diminishing property of the Laplace transform.

8 pages, 1 figure, tentatively accepted by adv. in appl. math