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Confirming Two Conjectures of Su and Wang
Two conjectures of Su and Wang (2008) concerning binomial coefficients are
proved. For and , we show that the finite sequence
is a P\'{o}lya frequency sequence. For
and , we show that there exists an integer such that the
infinite sequence , is log-concave for and log-convex for . 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.Comment: 8 pages, 1 figure, tentatively accepted by adv. in appl. mat
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