Following Boros-Moll, a sequence (an) is m-log-concave if Lj(an)⩾0\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}^{j}(a_{n})\geqslant0}$$\end{document} for all j = 0, 1, . . . , m. Here, L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document} is the operator defined by L(an)=an2-an-1an+1\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}$$\end{document}. By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality ak2⩾rak-1ak+1\documentclass[12pt]{minimal}
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\begin{document}$${a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}$$\end{document} 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 L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document} and are therefore ∞-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document} and L2\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}^{2}}$$\end{document} 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.
