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Borwein Conjectures

The Borwein conjectures concern signs of coefficients in finite products of factors from q-series. Here m and n are nonnegative integers. Use the definition of the q-series

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

and, for integers N>=0 and 0<=M<=N, define

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

This q-binomial coefficient is defined to be zero when M<0 or M>N. The first three Borwein conjectures (Bressoud 1996) state 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. Andrews' generalization (Bressoud 1996) states that (4) for m>=1, the polynomials A^|(m,n,t,q), B^|(m,n,t,q), and C^|(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)^(2n)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. Stanton's generalization (Bressoud 1996) uses the expansion (5) for integers k and a with k odd and 1<=a<k/2

(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)

The conjecture states that if a is relatively prime to k and m=n, the coefficients of F_nu(q) are nonnegative. Finally, Bressoud (1996) uses (6), for positive integers alpha, beta, and K satisfying alpha+beta<2K and -K+beta<=n-m<=K-alpha, to define

G(alpha,beta,K;q)=sum_(j)(-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. The sixth conjecture states that 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.

The first conjecture was proved by Wang (2022), and the second by Wang and Krattenthaler (2022). The Berkovich-Dhar conjecture concerns related powers of P_n(q)=product_(j=1)^(n)(1-q^(3j-2))(1-q^(3j-1)). Jiang (2026b) reported a proof of the related Berkovich-Dhar conjecture.

See also

Berkovich-Dhar Conjecture, q-Series

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References

Andrews, G. E.; Baxter, R. J.; Bressoud, D. M.; Burge, W. H.; Forrester, P. J.; and Viennot, G. "Partitions with Prescribed Hook Differences." Europ. J. Combin. 8, 341-350, 1987. https://doi.org/10.1016/S0195-6698(87)80041-0.Bressoud, D. M. "The Borwein Conjecture and Partitions with Prescribed Hook Differences." Elec. J. Combin. 3, No. 2, R4, 1996. https://doi.org/10.37236/1262.Jiang, S. "A Proof of the Berkovich-Dhar Conjecture Modulo Three." 7 Oct 2026b. https://arxiv.org/abs/2610.08496v2.Wang, C. "An Analytic Proof of the Borwein Conjecture." Adv. Math. 394, 108028, 2022. https://doi.org/10.1016/j.aim.2021.108028.Wang, C. and Krattenthaler, C. "An Asymptotic Approach to Borwein-Type Sign Pattern Theorems." 28 Jan 2022. https://arxiv.org/abs/2201.12415.

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Borwein Conjectures

Cite this as:

Weisstein, Eric W. "Borwein Conjectures." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BorweinConjectures.html

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