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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The reverse ultra log-concavity of the Boros-Moll polynomials


Authors: William Y. C. Chen and Cindy C. Y. Gu
Journal: Proc. Amer. Math. Soc. 137 (2009), 3991-3998
MSC (2000): Primary 05A20, 33F10
Published electronically: July 21, 2009
MathSciNet review: 2538559
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Abstract: We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence $ \{i!d_i(m)\}$ for any $ m\geq 2$, where $ d_i(m)$ are the coefficients of the Boros-Moll polynomials $ P_m(a)$. This inequality also leads to the fact that in the asymptotic sense, the Boros-Moll sequences are just on the borderline between ultra log-concavity and reverse ultra log-concavity. We propose two conjectures on the log-concavity and reverse ultra log-concavity of the sequence $ \{d_{i-1}(m) d_{i+1}(m)/d_i(m)^2\}$ for $ m\geq 2$.


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William Y. C. Chen
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: chen@nankai.edu.cn

Cindy C. Y. Gu
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: guchunyan@cfc.nankai.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09976-6
PII: S 0002-9939(09)09976-6
Keywords: Log-concavity, reverse ultra log-concavity, Boros-Moll polynomials.
Received by editor(s): August 31, 2008
Received by editor(s) in revised form: March 23, 2009
Published electronically: July 21, 2009
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.