9 problems

Let $k,n$ be positive integers, set $N=2n+1$ and $p=1$, and let $C_{ab}$ be the Cartan matrix of type $\textup{A}_{N-1}$. For nonnegative integers $m_i^{(a)}$, define the associate…

Let $q$ be a formal variable, and for $n\geq0$ let $(q^4;q^4)_n$ and $(q^m;q^m)_\infty$ denote the usual finite and infinite $q$-Pochhammer products; products with several argument…

For $|q|<1$, let $(a;q)_n$ denote the $q$-Pochhammer symbol, defined by $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$ and $(a;q)_\infty=\prod_{k=0}^{\infty}(1-aq^k)$; for multiple parameters…

Let $a,b,k$ be integers such that $0\leqslant a,b\leqslant k$, let $\tau\in\{-1,0,1\}$, and set … For weakly decreasing nonnegative integer sequences…

Let $(a;q)_n$ denote the $q$-Pochhammer symbol, and let $(a_1,a_2;q)_\infty=(a_1;q)_\infty(a_2;q)_\infty$. The variables $m,n$ range over nonnegative integers. The Andrews–Uncu con…

Non-negative coefficient conjecture. For any profile $c=(c_1,c_2,c_3)$, the polynomial $P_{n,c}(q)$ has non-negative coefficients. Moreover, if $\ell\not\equiv 0\pmod{3}$, then

Let $q$ and $x$ be formal variables, and for each fixed integer $m\geq1$ let $n_{m,k}$ be positive integers indexed by finitely many $k\in I_m$. Refined Rogers–Ramanujan conjecture…

Let $\tau\in\mathbb{Z}$ be a framing, let $Z^{(\mathbb{C}^3,\mathcal{D}_\tau)}(\mathbf{x};q)$ be the framed open-string partition function, and let $f_n^\tau(q)$ be the coefficient…

Let $m,n\geq 1$. Let $h_{\lambda}^{(m)}(w,z;q)$ denote the modified Rogers–Szegő factor defined earlier, let $P'_{\lambda}$ be the modified Hall–Littlewood polynomial, and let…