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 .

See also q -Series
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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 . Referenced
on Wolfram|Alpha Borwein Conjectures
Cite this as:
Weisstein, Eric W. "Borwein Conjectures."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/BorweinConjectures.html

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