Use the definition of the qSeries

(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.
See also qSeries
References
Andrews, G. E. et al. ``Partitions with Prescribed Hook Differences.'' Europ. J. Combin. 8, 341350, 1987.
Bressoud, D. M. ``The Borwein Conjecture and Partitions with Prescribed Hook Differences.'' Electronic J. Combinatorics 3, No. 2, R4, 114, 1996.
http://www.combinatorics.org/Volume_3/volume3_2.html#R4.
© 19969 Eric W. Weisstein
19990526