The Ramanujan Journal

, Volume 48, Issue 1, pp 117–129 | Cite as

r-log-concavity of partition functions

  • Qing-Hu Hou
  • Zuo-Ru ZhangEmail author


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.


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

Mathematics Subject Classification

05A17 11N37 65G99 



We would like to thank the referees for valuable comments.


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Copyright information

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Authors and Affiliations

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

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