Proof of Moll's Minimum Conjecture
William Y. C. Chen and Ernest X. W. Xia
Abstract: Let d_{i}(m) denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence {i(i+1)(d_{i}^{2}(m)-d_{i-1}(m)d_{i+1}(m))}_{1≤ i≤ m} attains its minimum at i = m with 2^{-2m}m(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 {d_{i}(m)}_{0≤ i≤ m}, and the log-concavity of the sequence {i!d_{i}(m)}_{0≤ i≤ m}. AMS Classification: 05A20; 11B83; 33F99 Keywords: ratio monotonicity, log-concavity, Boros-Moll polynomials Download: PDF |