Annals of Combinatorics

, Volume 20, Issue 1, pp 125–138 | Cite as

On Multiple and Infinite Log-Concavity

  • Luis A. Medina
  • Armin Straub


Following Boros-Moll, a sequence (a n ) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j =  0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality \({a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}\) 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 \({\mathcal{L}}\) and are therefore ∞-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 ∞-log-concave if convoluted with themselves a finite number of times.


log-concavity linear recurrences convolution 

Mathematics Subject Classification

05A20 39B12 


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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Puerto RicoSan JuanUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUnited States
  3. 3.Max-Planck-Institut für MathematikBonnGermany

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