Experiments with a Positivity Preserving Operator

TL;DR

This paper explores an operator that preserves positivity in series coefficients and applies it to rational functions, resulting in new functions whose positivity could confirm the original series' positivity, advancing understanding of positivity in multivariate functions.

Contribution

It introduces a positivity-preserving operator and demonstrates its application to rational functions, providing a method to generate functions with conjectured positive series coefficients.

Findings

New rational functions with positive coefficients are obtained.

The positivity of these functions implies the positivity of the original series.

The approach suggests a pathway to prove positivity conjectures in multivariate series.

Abstract

We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply positivity of the original series, and which, in a certain sense, cannot be improved any further.