论文标题
用于borwein型标志模式定理的渐近方法
An asymptotic approach to Borwein-type sign pattern theorems
论文作者
论文摘要
著名的（第一个）Borwein的猜想预测，对于所有积极整数〜$ n $，``borwein多项式''$ q^2）（1- q^2）（1- q^2）（1- q^4）（1-q^5）（1-q^5）（1-q^5）（1- q^5） $+ - + - \ cdots $。 [adv。数学。 394（2022），第108028号论文]。在本文中，我们从前面的论文中提取必需品，并将其提高到概念方法，以证明``Borwein like''标志模式语句。特别是，我们提供了原始的（第一个）Borwein猜想的新证明，这是第二个Borwein猜想的证明（预测``Borwein polytomial''正方形的标志模式也是$+ - + - \ cdots $），并且是``cdots $''的一部分证明，并且是``cuborwe''cuborwein'''cuborwein'' ``borwein多项式'）。讨论了许多进一步的申请。
The celebrated (First) Borwein Conjecture predicts that for all positive integers~$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.
