Confirming Two Conjectures of Su and Wang

TL;DR

This paper proves two conjectures by Su and Wang related to binomial coefficients, demonstrating their properties as Pólya frequency sequences and analyzing their log-concavity and log-convexity under certain conditions.

Contribution

It establishes the positivity and sequence properties of binomial coefficients in new parameter regimes, using methods from total positivity and Laplace transform theory.

Findings

Finite binomial coefficient sequences are Pólya frequency sequences.

Infinite binomial coefficient sequences exhibit log-concavity and log-convexity properties.

The proofs connect total positivity, planar networks, and variation-diminishing properties.

Abstract

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.