An asymptotic approach to Borwein-type sign pattern theorems

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.