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 {P_{m}(x)}_{m≥0}, where P_{m}(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 P_{m}(x) interlace the ratios of consecutive coefficients of P_{m+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 Download: PDF |