An asymptotic approach to Borwein-type sign pattern theorems

TL;DR

This paper introduces an asymptotic approach to prove Borwein-type sign pattern theorems, providing new proofs for existing conjectures and partial results for related polynomial powers, advancing understanding of their coefficient sign behaviors.

Contribution

It offers a new conceptual proof framework for Borwein conjectures and extends results to polynomial squares and cubes, broadening the scope of sign pattern analysis.

Findings

New proof of the First Borwein Conjecture

Proof of the Second Borwein Conjecture for polynomial squares

Partial proof of the cubic Borwein Conjecture

Abstract

The celebrated (First) Borwein Conjecture predicts that for all positive integers~$n$ the sign pattern of the coefficients of the ``Borwein polynomial'' $$(1-q)(1-q^2)(1-q^4)(1-q^5) \cdots(1-q^{3n-2})(1-q^{3n-1})$$ is $+--+--\cdots$. It was proved by the first author in [Adv. Math. 394 (2022), Paper No. 108028]. In the present paper, we extract the essentials from the former paper and enhance them to a conceptual approach for the proof of ``Borwein-like'' sign pattern statements. In particular, we provide a new proof of the original (First) Borwein Conjecture, a proof of the Second Borwein Conjecture (predicting that the sign pattern of the square of the ``Borwein polynomial'' is also $+--+--\cdots$), and a partial proof of a ``cubic'' Borwein Conjecture due to the first author (predicting the same sign pattern for the cube of the ``Borwein polynomial''). Many further applications are discussed.