Iterated sequences and the geometry of zeros

TL;DR

This paper investigates how certain nonlinear transformations affect the zeros of generating functions, characterizing those that preserve real non-positive zeros, confirming conjectures, and exploring implications for log-concavity and entire functions.

Contribution

It provides a characterization of transformations preserving real non-positive zeros, confirming several conjectures and extending results to entire functions in the Laguerre-Pólya class.

Findings

Transformations preserving real non-positive zeros are characterized.

Polynomials with only real non-positive zeros have infinitely log-concave coefficients.

Results extend to entire functions in the Laguerre-Pólya class.

Abstract

We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of P\'olya--Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial $a_0+a_1z +\cdots+a_nz^n$ has only real and non-positive zeros, then so does the polynomial $a_0^2+ (a_1^2-a_0a_2)z+...+ (a_{n-1}^2-a_{n-2}a_n)z^{n-1}+a_n^2z^n$. This confirms a conjecture of Fisk, McNamara-Sagan and Stanley, respectively. A consequence is that if a polynomial has only real and non-positive zeros, then its Taylor coefficients form an infinitely log-concave sequence. We extend the results to transcendental entire functions in the Laguerre-P\'olya class, and discuss the consequences to problems on iterated Tur\'an inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.