Use the definition of the q-series
(1)
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and define
(2)
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Then P. Borwein has conjectured that (1) the polynomials , , and defined by
(3)
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have nonnegative coefficients, (2) the polynomials , , and defined by
(4)
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have nonnegative coefficients, (3) the polynomials , , , , and defined by
(5)
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have nonnegative coefficients, (4) the polynomials , , and defined by
(6)
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have nonnegative coefficients, (5) for odd and , consider the expansion
(7)
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with
(8)
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then if is relatively prime to and , the coefficients of are nonnegative, and (6) given and , consider
(9)
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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.