Abstract:  We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence {i!d_i(m)} for any m ≥ 2, where d_i(m) are the coefficients of the Boros-Moll polynomials P_m(a). This inequality also leads to the fact that in the asymptotic sense, the Boros-Moll sequences are just on the borderline between ultra log-concavity and reverse ultra log-concavity. We propose two conjectures on the log-concavity and reverse ultra log-concavity of the sequence {d_i-1(m) d_i+1(m)/d_i (m)²} for m ≥ 2.

  AMS Classification:  05A20; 33F10

  Keywords:  log-concavity, reverse ultra log-concavity, Boros-Moll polynomials

  Download:   pdf