## 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-*q*^{2})(1-*q*^{4})(1-*q*^{5})
...(1-*q*^{3n-2})(1-*q*^{3n-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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