Vol. 254, No. 1, 2011

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Interlacing log-concavity of the Boros–Moll polynomials

William Y. C. Chen, Larry X. W. Wang and Ernest X. W. Xia

Vol. 254 (2011), No. 1, 89–99
Abstract

We say a sequence {Pm(x)}m0 of polynomials of degree m with positive coefficients is interlacingly log-concave if the ratios of consecutive coefficients of Pm(x) interlace the ratios of consecutive coefficients of Pm+1(x) for any m 0. Interlacing log-concavity of a sequence of polynomials is stronger than log-concavity of the polynomials themselves. We show that the Boros–Moll polynomials are interlacingly log-concave. Furthermore, we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacingly log-concave.

Keywords
interlacing log-concavity, log-concavity, Boros–Moll polynomial
Mathematical Subject Classification 2000
Primary: 05A20
Secondary: 33F10
Milestones
Received: 9 August 2010
Revised: 12 May 2011
Accepted: 17 May 2011
Published: 7 February 2012
Authors
William Y. C. Chen
Center for Combinatorics, LPMC-TJKLC
Nankai University
Tianjin 300071
China
Larry X. W. Wang
Center for Combinatorics, LPMC-TJKLC
Nankai University
Tianjin 300071
China
Ernest X. W. Xia
Center for Combinatorics, LPMC-TJKLC
Nankai University
Tianjin 300071
China