Infinite log-concavity for polynomial Pólya frequency sequences

Petter Brändén; Matthew Chasse

paper

Infinite log-concavity for polynomial Pólya frequency sequences

arXiv:1405.6378

Abstract

McNamara and Sagan conjectured that if $a_{0}, a_{1}, a_{2}, \dots$ is a Pólya frequency (PF) sequence, then so is $a_{0}^{2}, a_{1}^{2} - a_{0} a_{2}, a_{2}^{2} - a_{1} a_{3}, \dots$ . We prove this conjecture for a natural class of PF-sequences which are interpolated by polynomials. In particular, this proves that the columns of Pascal's triangle are infinitely log-concave, as conjectured by McNamara and Sagan. We also give counterexamples to the first mentioned conjecture. Our methods provide families of nonlinear operators that preserve the property of having only real and non-positive zeros.

12 pages