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# 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.

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