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.

The following versions are available:

Back to Christian Krattenthaler's home page.