Confirming two conjectures of Su and Wang on binomial coefficients

AbstractTwo 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+jak+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+jak+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