Brändén's Conjectures on the Boros-Moll Polynomials
William Y.C. Chen, Donna Q.J. Dou and Arthur L.B. Yang
Abstract: We prove two conjectures of Brändén on the real-rootedness of polynomials Qn(x) and Rn(x) which are related to the Boros-Moll polynomials Pn(x). In fact, we show that both Qn(x) and Rn(x) form Sturm sequences. The first conjecture implies the 2-log-concavity of Pn(x), and the second conjecture implies the 3-log-concavity of Pn(x).
AMS Classification: 26C10, 05A20, 30C15
Keywords: Boros-Moll Polynomials, Real-rootedness, Sturm sequence, 3-log-concavity