The $p=1$ case of the Feigin–Stoyanovsky identity

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 associated integers $M_i^{(a)}$ as in the Feigin–Stoyanovsky identity. Then the identity

$$\sum \frac{q^{\frac{1}{2}\sum_{a,b=1}^{N-1}\sum_{i=1}^{k-1} C_{ab}M_i^{(a)}M_i^{(b)}+\sum_{a=1}^{N-1}\sum_{i=1}^{k-1}(-1)^aM_i^{(a)}}}{\prod_{a=1}^{N-1}\prod_{i=1}^{k-1}(q)_{m_i^{(a)}}} =\frac{1}{(q)_\infty^{2n^2+n}} \sum \chi(\boldsymbol{v}/\boldsymbol{\rho}^{\ast})q^{\frac{\lVert\boldsymbol{v}\rVert^2-\lVert\boldsymbol{\rho}^{\ast}+k-1\rVert^2}{4(k+n)}}$$

holds, where $\boldsymbol{\rho}^{\ast}=(1,2,\ldots,n)$, $\chi(\boldsymbol{v}/\boldsymbol{w})=\prod_{i=1}^n\frac{v_i}{w_i}\prod_{1\leq i<j\leq n}\frac{v_i^2-v_j^2}{w_i^2-w_j^2}$, and the sum is over $\boldsymbol{v}\in\mathbb{Z}^n$ satisfying $v_i\equiv\rho_i^{\ast}+k-1\pmod{2k+2n}$. This is the specialization $p=1$ of the Feigin–Stoyanovsky identity; the surrounding discussion relates it to Macdonald's $C_n^{(1)}$ eta-function identity and, for $n=1$, to Jacobi's identity.