Annals of Combinatorics

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

On Multiple and Infinite Log-Concavity

Article
  • 93 Downloads

Abstract

Following Boros-Moll, a sequence (an) 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.

Keywords

log-concavity linear recurrences convolution 

Mathematics Subject Classification

05A20 39B12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahls, P., Devitt-Ryder, R., Nguyen, T.: The location of roots of logarithmically concave polynomials. Preprint (2011)Google Scholar
  2. 2.
    Boros, G., Moll, V.: Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge (2004)Google Scholar
  3. 3.
    Brändén P.: Iterated sequences and the geometry of zeros. J. Reine Angew. Math. 2011(658), 115–131 (2011)CrossRefGoogle Scholar
  4. 4.
    Chen, W.Y.C., Yang, A.L.B., Zhou, E.L.F.: Ratio monotonicity of polynomials derived from nondecreasing sequences. Electron. J. Combin. 17(1), #N37 (2010)Google Scholar
  5. 5.
    Craven, T., Csordas, G.: Iterated Laguerre and Turán inequalities. JIPAM. J. Inequal. Pure Appl. Math. 3(3), Art.39 (2002)Google Scholar
  6. 6.
    Csordas, G.: Iterated Turán inequalities and a conjecture of P. Brändén. In: Brändén, P., Passare, M., Putinar, M. (eds.) Notions of Positivity and the Geometry of Polynomials, pp. 103–113. Springer, Basel (2011)Google Scholar
  7. 7.
    Fisk, S.: Questions about determinants and polynomials. Preprint. Available online at: http://arxiv.org/abs/0808.1850 (2008)
  8. 8.
    Grabarek, L.: Non-Linear Coefficient-Wise Stability and Hyperbolicity Preserving Transformations. PhD thesis. University of Hawai‘i at Mānoa, Ann Arbor, MI (2012)Google Scholar
  9. 9.
    Hoggar S.G.: Chromatic polynomials and logarithmic concavity. J. Combin. Theory Ser. B 16, 248–254 (1974)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Johnson O., Goldschmidt C.: Preservation of log-concavity on summation. ESAIM Probab. Statist. 10, 206–215 (2006)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Katkova O.M., Vishnyakova A.M.: A sufficient condition for a polynomial to be stable. J. Math. Anal. Appl. 347(1), 81–89 (2008)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Kauers M., Paule P.: A computer proof of Moll’s log-concavity conjecture. Proc. Amer. Math. Soc. 135(12), 3847–3856 (2007)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Kurtz D.C.: A sufficient condition for all the roots of a polynomial to be real. Amer. Math. Monthly 99(3), 259–263 (1992)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    McNamara, P.R.W., Sagan, B.E.: Infinite log-concavity: developments and conjectures. In: Krattenthaler, C., Strehl, V., Kauers, M. (eds.) 21st International Conference on Formal Power Series and Algebraic Combinatorics, pp. 635–646. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2009)Google Scholar
  15. 15.
    Niculescu, C.P.: A new look at Newton’s inequalities. JIPAM. J. Inequal. Pure Appl. Math. 1(2), Art. 17 (2000)Google Scholar
  16. 16.
    Stanley, R.P.: Log-concave and unimodal sequences in algebra, combinatorics and geometry. In: Capobianco, M.F., Guan, M.G., Hsu, D.F., Tian, F. (eds.) Graph Theory and Its Applications: East and West, pp. 500–535. New York Academy of Sciences, New York (1989)Google Scholar
  17. 17.
    The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org
  18. 18.
    Zeilberger D.: A holonomic systems approach to special function identities. J. Comput. Appl. Math. 32(3), 321–368 (1990)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Zeilberger D.: The C-finite ansatz. Ramanujan J. 31(1), 23–32 (2013)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations