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Borwein Conjectures

Use the definition of the q-series

(a;q)_n=product_(j=0)^(n-1)(1-aq^j)
(1)

and define

[N; M]=((q^(N-M+1);q)_M)/((q;q)_m).
(2)

Then P. Borwein has conjectured that (1) the polynomials A_n(q), B_n(q), and C_n(q) defined by

(q;q^3)_n(q^2;q^3)_n=A_n(q^3)-qB_n(q^3)-q^2C_n(q^3)
(3)

have nonnegative coefficients, (2) the polynomials A_n^*(q), B_n^*(q), and C_n^*(q) defined by

(q;q^3)_n^2(q^2;q^3)_n^2=A_n^*(q^3)-qB_n^*(q^3)-q^2C_n^*(q^3)
(4)

have nonnegative coefficients, (3) the polynomials A_n^*(q), B_n^*(q), C_n^*(q), D_n^*(q), and E_n^*(q) defined by

(q;q^5)_n(q^2;q^5)_n(q^3;q^5)_n(q^4;q^5)_n= A_n^*(q^5)-qB_n^*(q^5)-q^2C_n^*(q^5)-q^3D_n^*(q^5)-q^4E_n^*(q^5)
(5)

have nonnegative coefficients, (4) the polynomials A_n^|(m,n,t,q), B_n^|(m,n,t,q), and C_n^|(m,n,t,q) 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)]
(6)

have nonnegative coefficients, (5) for k odd and 1<=a<=k/2, consider the expansion

(q^a;q^k)_m(q^(k-a);q^k)_n=sum_(nu=(1-k)/2)^((k-1)/2)(-1)^nuq^(k(nu^2+nu)/2-anu)F_nu(q^k)
(7)

with

F_nu(q)=sum_(j=-infty)^infty(-1)^jq^(j(k^2j+2knu+k-2a)/2)[m+n; m+nu+kj],
(8)

then if a is relatively prime to k and m=n, the coefficients of F_nu(q) are nonnegative, and (6) given alpha+beta<2K and -K+beta<=n-m<=K-alpha, consider

G(alpha,beta,K;q)=sum_(q)(-1)^jq^(j[K(alpha+beta)j+K(alpha+beta)]/2)[m+n; m+Kj],
(9)

the generating function for partitions inside an m×n rectangle with hook difference conditions specified by alpha, beta, and K. Let alpha and beta be positive rational numbers and k>1 an integer such that alphak and betak are integers. then if 1<=alpha+beta<=2k-1 (with strict inequalities for k=2) and -k+beta<=n-m<=k-alpha, then g(alpha,beta,k;q) 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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