Following Boros-Moll, a sequence \((a_n)\) is \(m\)-log-concave if
\(\mathcal{L}^j(a_n) \geq 0\) for all \(j = 0, 1, \cdots, m\). 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 Branden 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.
