Borwein Conjectures
المؤلف:
Andrews, G. E.
المصدر:
"Partitions with Prescribed Hook Differences." Europ. J. Combin. 8
الجزء والصفحة:
...
21-8-2019
2056
Borwein Conjectures
Use the definition of the q-series
 |
(1)
|
and define
![[N; M]=((q^(N-M+1);q)_M)/((q;q)_m).](http://mathworld.wolfram.com/images/equations/BorweinConjectures/NumberedEquation2.gif) |
(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
![(q;q^3)_m(q^2;q^3)_m(zq;q^3)_n(zq^2;q^3)_n
=sum_(t=0)^(2m)z^t[A^|(m,n,t,q^3)-qB^|(m,n,t,q^3)-q^2C^|(m,n,t,q^3)]](http://mathworld.wolfram.com/images/equations/BorweinConjectures/NumberedEquation6.gif) |
(6)
|
have nonnegative coefficients, (5) for 
odd and 
, consider the expansion
 |
(7)
|
with
![F_nu(q)=sum_(j=-infty)^infty(-1)^jq^(j(k^2j+2knu+k-2a)/2)[m+n; m+nu+kj],](http://mathworld.wolfram.com/images/equations/BorweinConjectures/NumberedEquation8.gif) |
(8)
|
then if 
is relatively prime to 
and 
, the coefficients of 
are nonnegative, and (6) given 
and 
, consider
![G(alpha,beta,K;q)=sum_(q)(-1)^jq^(j[K(alpha+beta)j+K(alpha+beta)]/2)[m+n; m+Kj],](http://mathworld.wolfram.com/images/equations/BorweinConjectures/NumberedEquation9.gif) |
(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.
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.
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