Branden's Conjectures on the Boros-Moll Polynomials

William Y. C. Chen; Donna Q. J. Dou; Arthur L. B. Yang

paper

Branden's Conjectures on the Boros-Moll Polynomials

arXiv:1205.0305

Abstract

We prove two conjectures of Brändén on the real-rootedness of polynomials $Q_{n} (x)$ and $R_{n} (x)$ which are related to the Boros-Moll polynomials $P_{n} (x)$ . In fact, we show that both $Q_{n} (x)$ and $R_{n} (x)$ form Sturm sequences. The first conjecture implies the 2-log-concavity of $P_{n} (x)$ , and the second conjecture implies the 3-log-concavity of $P_{n} (x)$ .

8 pages