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

DOI:
https://doi.org/10.1090/S0002-9939-09-09976-6

Published electronically:
July 21, 2009

MathSciNet review:
2538559

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Abstract | References | Similar Articles | Additional Information

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 for any , where are the coefficients of the Boros-Moll polynomials . 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 for .

**1.**G. Boros and V.H. Moll, A sequence of unimodal polynomials, J. Math. Anal. Appl. 237 (1999), 272-287. MR**1708173 (2000m:33007)****2.**G. Boros and V.H. Moll, The double square root, Jacobi polynomials and Ramanujan's master theorem, J. Comput. Appl. Math. 130 (2001), 337-344. MR**1827991 (2002d:33030)****3.**G. Boros and V.H. Moll, Irresistible Integrals, Cambridge University Press, Cambridge, 2004. MR**2070237 (2005b:00001)****4.**F. Brenti, Unimodal, log-concave, and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 413 (1989), 1-106. MR**963833 (90d:05014)****5.**W.Y.C. Chen and E.X.W. Xia, The ratio monotonicity of the Boros-Moll polynomials, Math. Comput., to appear.**6.**H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, European J. Combin. 29 (2008), 1544-1554. MR**2431746****7.**M. Kausers and P. Paule, A computer proof of Moll's log-concavity conjecture, Proc. Amer. Math. Soc. 135 (2007), 3847-3856. MR**2341935 (2009d:33063)****8.**T.M. Liggett, Ultra logconcave sequences and negative dependence, J. Combin. Theory. Ser. A 79 (1997), 315-325. MR**1462561 (98j:60018)****9.**V.H. Moll, The evaluation of integrals: A personal story, Notices Amer. Math. Soc. 49 (2002), 311-317. MR**1879857 (2002m:11105)****10.**V.H. Moll, Combinatorial sequences arising from a rational integral, Online Journal of Analytic Combin., No. 2 (2007), Art. 4. MR**2289956 (2008m:05006)****11.**R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph Theory and Its Applications: East and West, Ann. New York Acad. Sci. 576 (1989), 500-535. MR**1110850 (92e:05124)**

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Additional Information

**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:
https://doi.org/10.1090/S0002-9939-09-09976-6

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.