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 a_i(m). This sequence of polynomials is said to be interlacing 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 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.

  AMS Classification:  05A20; 33F10

  Keywords:  interlacing log-concavity, log-concavity, Boros-Moll polynomial.

  Download:   PDF