Confirming Two Conjectures of Su and Wang

Yaming Yu

Published 2009-01-04Version 1

Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For $n\geq k\geq 0$ and $b>a>0$, we show that the finite sequence $C_j=\binom{n+ja}{k+jb}$ is a P\'{o}lya frequency sequence. For $n\geq k\geq 0$ and $a>b>0$, we show that there exists an integer $m\geq 0$ such that the infinite sequence $\binom{n+ja}{k+jb}, j=0, 1,...$, is log-concave for $0\leq j\leq m$ and log-convex for $j\geq m$. 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.