Proof of Moll's Minimum Conjecture

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

  Abstract:  Let di(m) denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence {i(i+1)(di2(m)-di-1(m)di+1(m))}1≤ i≤ m attains its minimum at i = m with 2-2mm(m+1). This conjecture is a stronger than the log-concavity conjecture of Moll proved by Kauers and Paule. We give a proof of Moll's conjecture by utilizing the spiral property of the sequence {di(m)}0≤ i≤ m, and the log-concavity of the sequence {i!di(m)}0≤ i≤ m.

  AMS Classification:  05A20; 11B83; 33F99

  Keywords:  ratio monotonicity, log-concavity, Boros-Moll polynomials

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