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 Br\"and\'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.
Categories: math.CO math.NT
