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- W2102492869 abstract "Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if $mathcal{L}^j (a_n) geq 0$ 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 $a_k^2 geq r a_{k - 1} a_{k + 1}$ for large enough $r$. On the other hand, a recent result of Brand'en shows that $infty$-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 $infty$-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 $infty$-log-concave if convoluted with themselves a finite number of times." @default.
- W2102492869 created "2016-06-24" @default.
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- W2102492869 date "2014-05-07" @default.
- W2102492869 modified "2023-10-01" @default.
- W2102492869 title "On multiple and infinite log-concavity" @default.
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