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Sun’s log-concavity conjecture on the Catalan–Larcombe–French sequence

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Abstract

Let {P n } n≥0 denote the Catalan–Larcombe–French sequence, which naturally came from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence \({\left\{ {\sqrt[n]{{{P_n}}}} \right\}_{n \geqslant 1}}\), which was originally conjectured by Z. W. Sun. We also obtain the strict log-concavity of the sequence \({\left\{ {\sqrt[n]{{{V_n}}}} \right\}_{n \geqslant 1}}\), where {V n } n≥0 is the Fennessey–Larcombe–French sequence arising from the series expansion of the complete elliptic integral of the second kind.

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References

  1. Catalan, E.: Sur les Nombres de Segner. Rend. Circ. Mat. Palermo., 1, 190–201 (1887)

    Article  MATH  Google Scholar 

  2. Chen, W. Y. C., Guo, J. J. F., Wang, L. X. W.: Infinitely log-monotonic combinatorial sequences. Adv. Appl. Math., 52, 99–120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jarvis, F., Verrill, H. A.: Supercongruences for the Catalan–Larcombe–French numbers. Ramanujan J., 22, 171–186 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Larcombe, P. J., French, D. R.: On the “other” Catalan numbers: a historical formulation re-examined. Congr. Numer., 143, 33–64 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Larcombe, P. J., French, D. R., Fennessey, E. J.: The Fennessey–Larcombe–French sequence {1, 8, 144, 2432, 40000, …}: formulation and asymptotic form. Congr. Numer., 158, 179–190 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Larcombe, P. J., French, D. R., Fennessey, E. J.: The Fennessey–Larcombe–French sequence {1, 8, 144, 2432, 40000, …}: a recursive formulation and prime factor decomposition. Congr. Numer., 160, 129–137 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Ribenboim, P.: The Little Book of Bigger Primes, Second Edition, Springer-Verlag, New York, 2004

  8. Rudin, W.: Principles of Mathematical Analysis, Third Edition, McGraw-Hill, Inc., New York, 1976

  9. Sasvári, Z.: Inequalities for Binomial coefficients. J. Math. Anal. Appl., 236, 223–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sun, Z. W.: Conjectures involving arithmetical sequences, in Numbers Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H. Li and J. Liu), Proc. 6th China–Japan Seminar (Shanghai, August 15–17, 2011), World Sci., Singapore, 2013, pp. 244–258. arXiv: 1208.2683v9

  11. Wang, Y., Zhu, B. X.: Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences. Sci. China Math., 57(11), 2429–2435 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xia, E. X. W., Yao, O. X. M.: A criterion for the log-convexity of combinatorial sequences. Electron. J. Combin., 20(4), #P3 (2013)

  13. Yang, A. L. B., Zhao, J. J. Y.: Log-concavity of the Fennessey–Larcombe–French Sequence, preprint, arXiv: 1503.02151v1

  14. Zhao, F.-Z.: The log-behavior of the Catalan–Larcombe–French sequence. Int. J. Number Theory, 10, 177–182 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, P. R. China

    James J. Y. Zhao

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  1. James J. Y. Zhao
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Correspondence to James J. Y. Zhao.

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Supported by the 863 Program and the National Science Foundation of China

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Zhao, J.J.Y. Sun’s log-concavity conjecture on the Catalan–Larcombe–French sequence. Acta. Math. Sin.-English Ser. 32, 553–558 (2016). https://doi.org/10.1007/s10114-016-5446-y

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  • DOI: https://doi.org/10.1007/s10114-016-5446-y

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