The finite companion hierarchy for Andrews–Gordon identities

Let $n,k,i$ be integers with $n obreak\geq 0$, $k\nobreak\geq 5$, and $k>i\nobreak\geq 1$. Set $\alpha_{ij}=\max\{j-i+1,0\}$. Let ${a\brack b}'_q$ denote the ordinary $q$-binomial coefficient, except that it is defined to be $1$ when $a<0$ and $b=0$.

Finite companion Andrews–Gordon conjecture.

$$\begin{aligned} \sum_{n_1\geq \dots\geq n_{k-1}\geq n_k=0} &q^{n_1^2+n_2^2+\dots+n_{k-1}^2+n_i+\dots+n_{k-1}} \\[-2pt] &\quad\times\prod_{j=1}^{k-1}{2n-2\sum_{l=1}^{j-1}n_l-n_j-n_{j+1}-2\alpha_{ij}\brack n_j-n_{j+1}}'_q \\ &=\sum_{r=-\infty}^{\infty}(-1)^r q^{\frac{r((2k+1)r+2k-2i+1)}{2}}{2n\brack n-\frac{(2k+1)r}{2}+(2k-2i+1)\frac{(-1)^r-1}{4}}_q. \end{aligned}$$

This extends the known finite identities for smaller values of $k$ and is part of the proposed companion hierarchy to the Foda–Quano polynomial refinement of the Andrews–Gordon identities. It remains unproved for the stated range of parameters.