## Borwein Conjectures

Use the definition of the q-Series

 (1)

and define
 (2)

Then P. Borwein has conjectured 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, (4) the Polynomials , , and defined by

 (6)
have Nonnegative Coefficients, (5) for Odd and , consider the expansion

 (7)

with

 (8)

then if is Relatively Prime to and , the Coefficients of are Nonnegative, and (6) given and , consider

 (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. Then if (with strict inequalities for ) and , then has Nonnegative Coefficients.

Andrews, G. E. et al. Partitions with Prescribed Hook Differences.'' Europ. J. Combin. 8, 341-350, 1987.
Bressoud, D. M. The Borwein Conjecture and Partitions with Prescribed Hook Differences.'' Electronic J. Combinatorics 3, No. 2, R4, 1-14, 1996. http://www.combinatorics.org/Volume_3/volume3_2.html#R4.