“Borwein Conjectures”的概念、定义、习题解题方法及翻译-数学辞典

单词

Borwein Conjectures

释义

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

anInteger such that

and

are integers. Then if

(with strictinequalities for

) and

, then

has NonnegativeCoefficients.

See also q-Series

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/volume3_2.html#R4.

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