Abstract
We present an infinite family of Borwein type + - - conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.
Highlights
The so-called Borwein conjectures, due to Peter Borwein, were popularized by Andrews [1]
Into a power series in q and the sign pattern displayed by the coefficients
June 2018, in a conference at Penn State celebrating Andrews’ 80th birthday, Chen Wang, a young Ph.D. student studying at the University of Vienna, announced that he has vanquished the first of the Borwein conjectures
Summary
The so-called Borwein conjectures, due to Peter Borwein (circa 1990), were popularized by Andrews [1]. The first of these concerns the expansion of finite products of the form (1 − q)(1 − q 2 )(1 − q 4 )(1 − q 5 )(1 − q 7 )(1 − q 8 ). They do not appear to be very far from these conjectures in form and content They are on different lines from other extensions of Borwein conjectures considered in [2,3,5,10,11,13,14].
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