Confirming two conjectures of Su and Wang on binomial coefficients

Abstract

Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For n ⩾ k ⩾ 0 and b > a > 0 , we show that the finite sequence C j = ( n + j a k + j b ) is a Pólya frequency sequence. For n ⩾ k ⩾ 0 and a > b > 0 , we show that there exists an integer m ⩾ 0 such that the infinite sequence ( n + j a k + j b ) , j = 0 , 1 , … , is log-concave for 0 ⩽ j ⩽ m and log-convex for j ⩾ 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.