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.

  AMS Classification:  05A20, 33F10

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

  Download:   pdf