Preserving log-concavity and generalized triangles

Abstract

We introduce generalized triangles called s‐triangles for s given positive integer, as a bi‐indexed sequence of non negative numbers {as(n,k)}0≤k≤ns satisfying as(n,k) = 0 for k<0 or k>ns. A such s‐triangle is LC‐positive if for each r, the sequence of polynomials ∑k = rnsas(n,k)qk is q‐log‐concave. We extend some results of Wang and Yeh, Log‐concavity and LC‐positivity, J. Combin. Theory Ser. A (2007), and show that if as(n,k) is LC‐positive then the log‐concavity of the sequence {xn} implies the log‐concavity of the sequence {zn} defined by zn = ∑k = 0nsas(n,k)xk. Applications related to ordinary multinomials are given.