Chen Wang
and
Christian Krattenthaler
An asymptotic approach to Borwein-type sign pattern theorems
(60 pages)
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-q2)(1-q4)(1-q5)
...(1-q3n-2)(1-q3n-1)
is + - - + - - .... 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
+ - - + - - ...), 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.
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