r-log-concavity of partition functions

Article
  • 8 Downloads

Abstract

Let \(\hat{\mathscr {L}}\) be the operator given by \(\hat{\mathscr {L}} \{a_n\}_{n \ge 0} = \{a_{n+1}^2 - a_{n} a_{n+2} \}_{n \ge 0}\). A sequence \(\{ a_n \}_{n \ge 0}\) is called asymptotically r-log-concave if \(\hat{\mathscr {L}}^k \{a_n\}_{n \ge N}\) are non-negative sequences for \(1 \le k \le r\) and some integer N. Let p(n) be the number of integer partitions of n. We prove that the sequence \(\{p(n)\}_{n \ge 1}\) is asymptotically r-log-concave for any positive integer r. Moreover, we give a method to compute the explicit N such that \(\{p(n)\}_{n \ge N}\) is r-log-concave.

Keywords

r-log-concavity Partition function Hardy–Ramanujan–Rademacher formula 

Mathematics Subject Classification

05A17 11N37 65G99 

Notes

Acknowledgements

We would like to thank the referees for valuable comments.

References

  1. 1.
    Chen, W.Y.C.: Recent developments on log-concavity and q-logconcavity of combinatorial polynomials. In: DMTCS Proceeding of 22nd International Conference on Formal Power Series and Algebraic Combinatorics (2010)Google Scholar
  2. 2.
    Chen, W.Y.C., Xia, E.X.W.: The \(2\)-log-convexity of the Apéry numbers. Proc. Am. Math. Soc. 139, 391–400 (2011)CrossRefMATHGoogle Scholar
  3. 3.
    Chen, W.Y.C., Zheng, K.Y.: The log-behavior of \(\root n \of {p(n)}\) and \(\root n \of {p(n)/n}\) (2015, arXiv preprint). arXiv:1511.02558
  4. 4.
    Chen, W.Y.C., Wang, L.X.W., Xie, G.Y.B.: Finite differences of the logarithm of the partition function. Math. Comput. 85(298), 825–847 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    DeSalvo, S., Pak, I.: Log-concavity of the partition function. Ramanujan J. 38, 61–73 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17, 75–175 (1918)CrossRefMATHGoogle Scholar
  7. 7.
    Hou, Q.H., Zhang, Z.R.: Asymptotic \(r\)-log-convexity and P-recursive sequence. arXiv:1609.07840
  8. 8.
    Lehmer, D.H.: On the series for the partition function. Trans. Am. Math. Soc. 43, 271–292 (1938)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lehmer, D.H.: On the remainders and convergence of the series for the partition function. Trans. Am. Math. Soc. 46, 362–373 (1939)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rademacher, H.: On the partition function \(p(n)\). Proc. Lond. Math. Soc. 2(1), 241–254 (1938)CrossRefMATHGoogle Scholar
  11. 11.
    Sun, Z.-W.: On a sequence involving sums of primes. Bull. Aust. Math. Soc. 88, 197–205 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China

Personalised recommendations