Log-concavity and LC-positivity

A triangle {a(n, k)}0≤k≤n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials ∑n k=r a(n, k)q k is q-log-concave. It is double LC-positive if both triangles {a(n, k)} and {a(n, n − k)} are LC-positive. We show that if {a(n, k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by zn = ∑n k=0 a(n, k)xk, and if {a(n, k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by zn = ∑n k=0 a(n, k)xkyn−k. Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence. MSC: 05A20; 15A04; 05A15; 15A48