The reverse ultra log-concavity of the Boros-Moll polynomials

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 ) } \{i!d_i(m)\} for any m ≥ 2 m\geq 2 , where d i ( m ) d_i(m) are the coefficients of the Boros-Moll polynomials P m ( a ) 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 } \{d_{i-1}(m) d_{i+1}(m)/d_i(m)^2\} for m ≥ 2 m\geq 2 .