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.

REFERENCES:

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/Abstracts/v3i2r4.html.