13 problems

Two-parameter Borwein positivity conjecture. The polynomials $A_{L,m}(q)$ for $0\leq m\leq L$ and $C_{L,m}(q)$ for $0\leq m\leq L-1$ have nonnegative coefficients.

Refined Borwein coefficient-positivity conjecture. The polynomials $\mathcal{A}_{L,M}(q)$ and $\mathcal{C}_{L,M}(q)$ have nonnegative coefficients.

For $k,n\in\mathbb{N}_0$, let $G_{k,n}(q)$ be defined by … The polynomials $z_{k,j}(q)$ are to be chosen for $0\le j\le n$. Partial-fraction positivity conjecture for $G_{k,n}(q)$.…

For $k,n\in\mathbb{N}$ with $k\ge n+1$, let $G_{k,n}(q)$ be defined by … Write $A(q)\succeq0$ when all coefficients of $A(q)$ are nonnegative. Monotonicity conjecture for…

For $k,n\in\mathbb{N}_0$, define $G_{k,n}(q)$ by … Here $\begin{bmatrix}a\b\end{bmatrix}_q$ denotes the $q$-binomial coefficient, and $A(q)\succeq0$ means that all coefficients of…

Chern's conjecture. This $q$-series has nonnegative coefficients in its expansion. The source presents this as a sufficient condition for the preceding double-series conjecture aft…

Chern's conjecture. This double series has nonnegative coefficients in its expansion. This would provide a uniform $q$-series proof of the parity-bias inequalities previously estab…

For nonnegative integers $A,B$, define the $q$-binomial coefficient by … Suppose that $M,N\in\mathbb Z^+$, that $\alpha$ and $\beta$ are positive rational numbers, and that $K$ is…

Third Borwein conjecture. Each of $v_n(q)$, $\phi_n(q)$, $\chi_n(q)$, $\psi_n(q)$, and $\omega_n(q)$ has non-negative coefficients. This is one of the sign-pattern conjectures for…

The generalized partial theta Borwein conjecture. For fixed $k$, if $m_1,m_2\geq1$ and $n_1,n_2,n_3$ are sufficiently large, then $A(q)$, $B(q)$, and $C(q)$ have non-negative coeff…

The first partial theta Borwein conjecture. For each $m,k\geq0$, there is a non-negative integer $N_{m,k}$ such that, if $n\geq N_{m,k}$, then $A_{m,n,k}(q)$, $B_{m,n,k}(q)$, and…

Diagonal-positivity conjecture. The series $\operatorname{diag}_F$ is nonnegative if and only if

Let $N_1,N_2,\ldots,N_k\ge2$ be integers, let $n$ be a positive integer, and let ${\bf q}_{\bf N}(z)$ be the function associated with the mirror map. For a power series, write…