The Reverse Ultra Log-Concavity of the Boros-Moll Polynomials

William Y.C. Chen and Cindy C.Y. Gu

  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!di(m)} for any m ≥ 2, where di(m) are the coefficients of the Boros-Moll polynomials Pm(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 {di-1(m)di+1(m)/di(m)2} for m ≥ 2.

  AMS Classification:  05A20; 33F10

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

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