3 problems

Constant-term classification problem. Can one classify all such integer sequences for which

Let $\mathcal{F}_n(\mathbf{x};\mathbf{a};q)$ denote the q-Dyson product, let $\sigma_n(\mathbf{a})=a_1+\cdots+a_n$, and let $[x_rx_s/(x_tx_u)]F$ denote the corresponding coefficien…

Let $n$ and $a_1,a_2,\dots,a_n$ be nonnegative integers, and let $q$ be an indeterminate. Consider the Laurent polynomial … Define $[m]_q=1+q+\cdots+q^{m-1}$ and…