Log-concavity and LC-positivity

Abstract

A triangle $\{a(n,k)\}_{0\le k\le n}$ of nonnegative numbers is LC-positive if for each $r$, the sequence of polynomials $\sum_{k=r}^{n}a(n,k)q^k$ 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 $z_n=\sum_{k=0}^{n}a(n,k)x_k$, 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 $z_n=\sum_{k=0}^{n}a(n,k)x_ky_{n-k}$. 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.Comment: 16 page

Duality and Asymptotics for a Class of Nonneutral Discrete Moran Models

“…Note that u is alternating but LC. The convolution of two LC sequences is LC (see [30] for instance). So ((α * u) i ) i∈{0,...,N} is LC and positive (because π i is positive).…”

Multimodality of the Markov Binomial Distribution

“…Inspired by the proof of Theorem 3.10 of [17], we use Lemma 4.1 to show the log-concavity of an important class of sequences. is log-concave.…”

Schur positivity and the q-log-convexity of the Narayana polynomials

“…The Davenport-Pólya theorem [9] states that if (a n ) n≥0 and (b n ) n≥0 are log-convex then their binomial convolution c n = n k=0 n k a k b n−k , n≥ 0 is also log-convex. It is known that the binomial convolution also preserves logconcavity [36]. Nevertheless, there exist log-convexity preserving transformations that do not preserve log-concavity, such as the componentwise sum [19].…”

“…Nevertheless, there exist log-convexity preserving transformations that do not preserve log-concavity, such as the componentwise sum [19]. There are also log-concavity preserving transformations that do not preserve log-convexity, such as the ordinary convolution [36].…”