Borwein Conjectures

Use the definition of the q-series


and define

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

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


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


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


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


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



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

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],

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


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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.

Referenced on Wolfram|Alpha

Borwein Conjectures

Cite this as:

Weisstein, Eric W. "Borwein Conjectures." From MathWorld--A Wolfram Web Resource.

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