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 ∑ k = r n 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 { x k } implies that of the sequence { z n } defined by z n = ∑ k = 0 n a ( n , k ) x k , and if { a ( n , k ) } is double LC-positive then the log-concavity of sequences { x k } and { y k } implies that of the sequence { z n } defined by z n = ∑ k = 0 n a ( n , k ) x k y n − 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.