516 research outputs found
Log-concavity and LC-positivity
A triangle of nonnegative numbers is LC-positive
if for each , the sequence of polynomials is
-log-concave. It is double LC-positive if both triangles and
are LC-positive. We show that if is LC-positive
then the log-concavity of the sequence implies that of the sequence
defined by , and if is
double LC-positive then the log-concavity of sequences and
implies that of the sequence defined by
. 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
Preservation of log-concavity on summation
We extend Hoggar's theorem that the sum of two independent discrete-valued
log-concave random variables is itself log-concave. We introduce conditions
under which the result still holds for dependent variables. We argue that these
conditions are natural by giving some applications. Firstly, we use our main
theorem to give simple proofs of the log-concavity of the Stirling numbers of
the second kind and of the Eulerian numbers. Secondly, we prove results
concerning the log-concavity of the sum of independent (not necessarily
log-concave) random variables
Recurrence Relations for Strongly q-Log-Convex Polynomials
We consider a class of strongly q-log-convex polynomials based on a
triangular recurrence relation with linear coefficients, and we show that the
Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the
Dowling polynomials are strongly q-log-convex. We also prove that the Bessel
transformation preserves log-convexity.Comment: 15 page
On unimodality problems in Pascal's triangle
Many sequences of binomial coefficients share various unimodality properties.
In this paper we consider the unimodality problem of a sequence of binomial
coefficients located in a ray or a transversal of the Pascal triangle. Our
results give in particular an affirmative answer to a conjecture of Belbachir
et al which asserts that such a sequence of binomial coefficients must be
unimodal. We also propose two more general conjectures.Comment: 12 pages, 2 figure
Infinite log-concavity: developments and conjectures
Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new
sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is
log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k)
"infinitely log-concave" if L^i(a_k) is nonnegative for all i >= 1. Boros and
Moll conjectured that the rows of Pascal's triangle are infinitely log-concave.
Using a computer and a stronger version of log-concavity, we prove their
conjecture for the nth row for all n <= 1450. We also use our methods to give a
simple proof of a recent result of Uminsky and Yeats about regions of infinite
log-concavity. We investigate related questions about the columns of Pascal's
triangle, q-analogues, symmetric functions, real-rooted polynomials, and
Toeplitz matrices. In addition, we offer several conjectures.Comment: 21 pages. Minor changes and additional references. Final version, to
appear in Advances in Applied Mathematic
- …