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# On multiple and infinite log-concavity

On multiple and infinite log-concavity
Luis A. Medina, Armin Straub — Annals of Combinatorics — Volume 20, Number 1, 2016, Pages 125-138

## Abstract

Following Boros--Moll, a sequence $$(a_n)$$ is $$m$$-log-concave if $$\mathcal{L}^j (a) \geqslant 0$$ for all $$j = 0, 1, \ldots, m$$. Here, $$\mathcal{L}$$ is the operator defined by $$\mathcal{L} (a_n) = a_n^2 - a_{n - 1} a_{n + 1}$$. By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is $$\infty$$-log-concave if it satisfies the stronger inequality $$a_k^2 \geqslant r a_{k - 1} a_{k + 1}$$ for large enough $$r$$. On the other hand, a recent result of Brändén shows that $$\infty$$-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this note, we investigate sequences, which are fixed by a power of the operator $$\mathcal{L}$$ and are therefore $$\infty$$-log-concave for a very different reason. Suprisingly, we find that sequences fixed by the non-linear operators $$\mathcal{L}$$ and $$\mathcal{L}^2$$ are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become $$\infty$$-log-concave if convoluted with themselves a finite number of times.

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## BibTeX

@article{infinitelogconcavity-2016,
author = {Luis A. Medina and Armin Straub},
title = {On multiple and infinite log-concavity},
journal = {Annals of Combinatorics},
year = {2016},
volume = {20},
number = {1},
pages = {125--138},
doi = {10.1007/s00026-015-0292-7},
}