Borwein Conjectures

The Borwein conjectures concern signs of coefficients in finite products of factors from q-series. Here and are nonnegative integers. Use the definition of the q-series

(1)

and, for integers and , define

(2)

This q-binomial coefficient is defined to be zero when or . The first three Borwein conjectures (Bressoud 1996) state that (1) the polynomials , , and defined by

(3)

have nonnegative coefficients, (2) the polynomials , , and defined by

(4)

have nonnegative coefficients, (3) the polynomials , , , , and defined by

(5)

have nonnegative coefficients. Andrews' generalization (Bressoud 1996) states that (4) for , the polynomials , , and defined by

(6)

have nonnegative coefficients. Stanton's generalization (Bressoud 1996) uses the expansion (5) for integers and with odd and

(7)

with

(8)

The conjecture states that if is relatively prime to and , the coefficients of are nonnegative. Finally, Bressoud (1996) uses (6), for positive integers , , and satisfying and , to define

(9)

the generating function for partitions inside an rectangle with hook difference conditions specified by , , and . Let and be positive rational numbers and an integer such that and are integers. The sixth conjecture states that if (with strict inequalities for ) and , then has nonnegative coefficients.

The first conjecture was proved by Wang (2022), and the second by Wang and Krattenthaler (2022). The Berkovich-Dhar conjecture concerns related powers of . Jiang (2026b) reported a proof of the sign assertions of the Berkovich-Dhar conjecture for all powers , 5, 6, 7, and 8. In each case, the coefficients are nonnegative, and the coefficients change sign exactly once, from positive to negative, after zero terms are omitted. The reported proof has not been independently verified here.