Annals of Combinatorics

, Volume 20, Issue 1, pp 125–138

# On Multiple and Infinite Log-Concavity

Article

## Abstract

Following Boros-Moll, a sequence (an) is m-log-concave if $${\mathcal{L}^{j}(a_{n})\geqslant0}$$ for all j =  0, 1, . . . , m. Here, $${\mathcal{L}}$$ is the operator defined by $${\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}$$. By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality $${a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}$$ for large enough r. On the other hand, a recent result of Brändén shows that ∞-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator $${\mathcal{L}}$$ and are therefore ∞-log-concave for a very different reason. Surprisingly, 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 ∞-log-concave if convoluted with themselves a finite number of times.

### Keywords

log-concavity linear recurrences convolution

05A20 39B12

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