Infinite log-concavity for polynomial P\'olya frequency sequences

TL;DR

This paper proves a conjecture about the infinite log-concavity of polynomial Pólya frequency sequences, including Pascal's triangle, and introduces methods for preserving real non-positive zeros.

Contribution

It establishes the infinite log-concavity for a class of PF-sequences interpolated by polynomials, confirming a conjecture for Pascal's triangle.

Findings

Proved the conjecture for polynomial-interpolated PF-sequences.

Demonstrated that Pascal's triangle columns are infinitely log-concave.

Provided counterexamples to a related conjecture.

Abstract

McNamara and Sagan conjectured that if $a_0,a_1, a_2, \ldots$ is a P\'olya frequency (PF) sequence, then so is $a_0^2, a_1^2 -a_0a_2, a_2^2-a_1a_3, \ldots$. 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.