22 problems

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…

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…

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…

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

Assume $|q|<1$ and $z\ne0$. Here $E(q):=\prod_{k=1}^{\infty}(1-q^k)$, $(z;q)_\infty:=\prod_{k=0}^{\infty}(1-zq^k)$, and $[z;q]_\infty:=(z;q)_\infty(q/z;q)_\infty$. The notation…

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.

Peter Borwein's conjecture. The polynomials $A_n(q)$, $B_n(q)$ and $C_n(q)$ 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…

Andrews–Bachraoui's positivity conjecture. The coefficients $c_{k,m}(n)$ are positive for $k,m\in\mathbb{N}$, including the stated case $k=2$, $m=4$. This concerns positivity of co…

For $k\geq 1$, define the finite sums … Here $(a,b,c;q)_\infty=\prod_{m=0}^{\infty}(1-aq^{b+mc})$, so $(q,q^4,q^5;q^5)_\infty$ and $(q^2,q^3,q^5;q^5)_\infty$ are the corresponding…

Let a ratio of $q$-factorials be expressed through its multiplicity data, with the associated inequalities comparing the multiplicities of the numerator and denominator factors. Wa…

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…

Positivity conjecture. The coefficients of $\mathscr{F}_t(1-q)$, $\mathscr{F}_t(\frac{1}{1+q})$, and $\mathscr{F}_t(\frac{1-q}{1+q})$ are positive for all $t\geq 1$; the coefficien…

Let $f_{2,n}(z)$ be the sparse polynomial sequence, and define $\mathcal L(a_n)=a_n^2-a_{n-1}a_{n+1}$ for a sequence $(a_n)$. For positive integers $k$, let $\mathcal L^k$ denote t…

For $m\geq3$, let … and define … where $\operatorname{PT}'_q$ is the positive-part operator used in the source. Write $[q^i]G_m(x,q)$ for the coefficient of $q^i$. Almost-positivit…

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…

For real $c$ and integer $d$, define … where $e_1$ and $e_d$ are the elementary symmetric polynomials in $x_1,\ldots,x_d$. Gillis–Reznick–Zeilberger's conjecture. For $d\geq 4$,…

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…