2-Log-concavity of the Boros-Moll Polynomials

William Y.C. Chen and Ernest X.W. Xia

  Abstract:  The Boros-Moll polynomials Pm(a) arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that Pm(a) is 2-log-concave for any m ≥ 2. Let di(m) be the coefficient of ai in Pm(a). We also show that the sequence {i(i+1) (d2i(m) - di-1 (m)di+1(m))}1≤i≤m is log-concave.

  AMS Classification:  05A10, 05A20; 33F10

  Keywords:  2-log-concavity, Boros-Moll polynomial, quartic integral

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