On multiple and infinite log-concavity

On multiple and infinite log-concavity
Luis A. Medina, Armin Straub — Annals of Combinatorics — Volume 20, Number 1, 2016, Pages 125-138


Following Boros--Moll, a sequence \((a_n)\) is \(m\)-log-concave if \(\mathcal{L}^j (a) \geqslant 0\) 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 \(a_k^2 \geqslant r a_{k - 1} a_{k + 1}\) for large enough \(r\). On the other hand, a recent result of Brändén 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.


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    author = {Luis A. Medina and Armin Straub},
    title = {On multiple and infinite log-concavity},
    journal = {Annals of Combinatorics},
    year = {2016},
    volume = {20},
    number = {1},
    pages = {125--138},
    doi = {10.1007/s00026-015-0292-7},