Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials

TL;DR

This paper investigates the positivity of Maclaurin coefficients in polynomials built from rising factorials and log-concave sequences, exploring their zeros, coefficients, and related Toeplitz determinants, supported by conjectures and numerical evidence.

Contribution

It introduces new positivity results for polynomials involving rising factorials and proposes several conjectures on their zeros and coefficients, extending the understanding of these mathematical structures.

Findings

Proved positivity of Maclaurin coefficients in certain polynomials

Formulated conjectures on zeros and coefficients of generalized polynomials

Supported conjectures with numerical evidence

Abstract

In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.