The Interlacing Log-concavity of the Boros-Moll Polynomials

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

  Abstract: We introduce the notion of interlacing log-concavity of a polynomial sequence {Pm(x)}m≥0, where Pm(x) is a polynomial of degree m with positive coefficients. This sequence is said to be 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. The interlacing log-concavity of a sequence of polynomials is stronger than the log-concavity of the polynomials themselves. We show that the Boros-Moll polynomials are interlacingly log-concave. Furthermore, we give a sufficient condition for the interlacing log-concavity which implies that some classical combinatorial polynomials are interlacingly log-concave.

  AMS Classification:  05A20; 33F10

  Keywords:  interlacing log-concavity, log-concavity, the Boros-Moll polynomials

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