Confirming Two Conjectures of Su and Wang

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 Cj = ( n+ja k+jb ) 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+ja k+jb ) , 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.