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

DOI: 10.2140/pjm.2011.254.89

Abstract

We say a sequence {P_m(x)}_m≥0 of polynomials of degree m with positive coefficients is interlacingly log-concave if the ratios of consecutive coefficients of P_m(x) interlace the ratios of consecutive coefficients of P_m+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