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Interlacing Log-concavity of the Boros-Moll Polynomials

Abstract

We introduce the notion of interlacing log-concavity of a polynomial sequence {Pm(x)}mβ‰₯0\{P_m(x)\}_{m\geq 0}, where Pm(x)P_m(x) is a polynomial of degree m with positive coefficients ai(m)a_{i}(m). This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of Pm(x)P_m(x) interlace the ratios of consecutive coefficients of Pm+1(x)P_{m+1}(x) for any mβ‰₯0m\geq 0. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.Comment: 10 page

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