The ratio monotonicity of the Boros-Moll polynomials

Authors:
William Y. C. Chen and Ernest X. W. Xia

Journal:
Math. Comp. **78** (2009), 2269-2282

MSC (2000):
Primary 05A20, 33F10

DOI:
https://doi.org/10.1090/S0025-5718-09-02223-6

Published electronically:
February 11, 2009

MathSciNet review:
2521289

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Abstract: In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the -log-concavity.

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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

**Ernest X. W. Xia**

Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China

Email:
xxw@cfc.nankai.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-09-02223-6

Keywords:
Ratio monotone property,
spiral property,
unimodality,
log-concavity,
Jacobi polynomials,
Boros-Moll polynomials.

Received by editor(s):
June 26, 2008

Received by editor(s) in revised form:
September 26, 2008

Published electronically:
February 11, 2009

Additional Notes:
This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.