Brändén's Conjectures on the Boros–Moll Polynomials

Abstract

We prove two conjectures of Brändén on the real-rootedness of the 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).