Skew log-concavity of the Boros-Moll sequences
- Eric H Liu^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-017-1394-z
© The Author(s) 2017
Received: 14 January 2017
Accepted: 2 May 2017
Published: 18 May 2017
Abstract
Let \(\{T(n,k)\}_{0\leq n < \infty, 0\leq k \leq n} \) be a triangular array of numbers. We say that \(T(n,k)\) is skew log-concave if for any fixed n, the sequence \(\{T(n+k,k)\}_{0 \leq k <\infty}\) is log-concave. In this paper, we show that the Boros-Moll sequences are almost skew log-concave.
Keywords
MSC
1 Introduction and main result
In this paper, we give a new definition, i.e., skew log-concavity. Let \(\{T(n,k)\}_{0\leq n < \infty, 0\leq k \leq n} \) be a triangular array of numbers. We say that \(T(n,k)\) is skew log-concave if for any fixed n, the sequence \(\{T(n+k,k)\}_{0 \leq k <\infty}\) is log-concave. We will show that the Boros-Moll sequences are almost skew log-concave.
The main results of this paper can be stated as follows.
Theorem 1.1
2 Proof of Theorem 1.1
From (1.4), we see that \(d_{m}(m)=2^{-m}{2m \choose m} \), which implies that (1.7) holds.
Declarations
Acknowledgements
This work was supported by the National Science Foundation of China (11526136, 11501356).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Boros, G, Moll, VH: A sequence of unimodal polynomials. J. Math. Anal. Appl. 237, 272-285 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Moll, VH: The evaluation of integrals: a personal story. Not. Am. Math. Soc. 49(3), 311-317 (2002) MathSciNetMATHGoogle Scholar
- Kausers, M, Paule, P: A computer proof of Moll’s log-concavity conjecture. Proc. Am. Math. Soc. 135(12), 3847-3856 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Chen, WYC, Xia, EXW: The ratio monotonicity of Boros-Moll polynomials. Math. Comput. 78, 2269-2282 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Chen, WYC, Xia, EXW: A proof of Moll’s minimum conjecture. Eur. J. Comb. 34, 787-791 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Chen, WYC, Wang, LXW, Xia, EXW: The interlacing log-concavity of Boros-Moll polynomials. Pac. J. Math. 254, 89-99 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Chen, WYC, Gu, CCY: The reverse ultra log-concavity of the Boros-Moll polynomials. Proc. Am. Math. Soc. 137, 3991-3998 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Chen, WYC, Xia, EXW: 2-log-concavity of the Boros-Moll polynomials. Proc. Edinb. Math. Soc. 56, 701-722 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Xia, EXW: The concavity and convexity of the Boros-Moll sequences. Electron. J. Comb. 22, Paper #P1.8 (2015) MathSciNetMATHGoogle Scholar