Borwein Conjectures

Use the definition of the q-Series

$$(a;q)_n\equiv \prod_{j=0}^{n-1} (1-a q^j)$$ (1)

and define

$$\left[{\matrix{N\cr M\cr}}\right] \equiv {(q^{N-M+1}; q)_M\over (q; q)_m}.$$ (2)

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

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

have Nonnegative Coefficients, (2) the Polynomials $A^*_n(q)$, $B^*_n(q)$, and $C^*_n(q)$ defined by

$$(q;q^3)^2_n(q^2;q^3)^2_n=A^*_n(q^3)-q B_n^*(q^3)-q^2 C_n^*(q^3)$$ (4)

have Nonnegative Coefficients, (3) the Polynomials $A^\star_n(q)$, $B^\star_n(q)$, $C^\star_n(q)$, $D^\star_n(q)$, and $E^\star_n(q)$ defined by

$(q;q^5)_n(q^2;q^5)_n(q^3;q^5)_n(q^4;q^5)_n=$
$A_n^\star(q^5)-q B_n^\star(q^5)-q^2 C_n^\star(q^5)-q^3 D_n^\star(q^5)-q^4 E_n^\star(q^5)\quad$	(5)

have Nonnegative Coefficients, (4) the Polynomials $A^\dagger_n(m,n,t,q)$, $B^\dagger_n(m,n,t,q)$, and $C^\dagger_n(m,n,t,q)$ defined by

$(q;q^3)_m(q^2;q^3)_m(zq;q^3)_n(zq^2;q^3)_n$
$=\sum_{t=0}^{2m} z^t[A^\dagger(m,n,t,q^3)-qB^\dagger(m,n,t,q^3)-q^2C^\dagger(m,n,t,q^3)]\quad$	(6)

have Nonnegative Coefficients, (5) for $k$ Odd and $1\leq a\leq k/2$, consider the expansion

$$(q^a;q^k)_m(q^{k-a};q^k)_n=\sum_{\nu=(1-k)/2}^{(k-1)/2}(-1)^\nu q^{k(\nu^2+\nu)/2-a\nu}F_\nu(q^k)$$ (7)

with

$$F_\nu(q)=\sum_{j=-\infty}^\infty (-1)^j q^{j(k^2j+2k\nu+k-2a)/2}\left[{\matrix{m+n\cr m+\nu+kj\cr}}\right],$$ (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 $-K+\beta\leq n-m\leq K-\alpha$, consider

$$G(\alpha,\beta,K;q)=\sum_q(-1)^j q^{j[K(\alpha+\beta)j+K(\alpha+\beta)]/2}\left[{\matrix{m+n\cr m+Kj\cr}}\right],$$ (9)

the Generating Function for partitions inside an $m\times n$ 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 $\alpha K$ and $\beta K$ are integers. Then if $1\leq \alpha+ \beta\leq 2K-1$ (with strict inequalities for $K=2$) and $-K+\beta\leq n-m\leq K-\alpha$, then $G(\alpha, \beta, K;q)$ has Nonnegative Coefficients.

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.