Generalized basic hypergeometric formula for separated Laurent polynomials

Let $n$ be a positive integer, let $g$ and $ell$ be parameters, and let $\lambda=(\lambda_1,\ldots,\lambda_n)$. Write $(y;q)_m$ for the $q$-Pochhammer symbol and ${}_n\varphi_{n-1}$ for the basic hypergeometric series. Define

$$a_j=\ell^{n-j+1}q^{\lambda_1-\lambda_{n-j+1}+1},\qquad b_j=a_j\ell^{-1}.$$

The generalized formula for the separated Laurent polynomial is the generalized formula for $S^{(\ell;q)}_\lambda(y)$.

$$S^{(\ell;q)}_{\lambda}(y)=y^{\lambda_1}(y;q)_{1-ng}\,{}_n\varphi_{n-1}\left[\begin{array}{c}a_1,\ldots,a_n\\ b_1,\ldots,b_{n-1}\end{array};q,y\right].$$

This extends the formula previously given for the $A_2$ separated polynomial to arbitrary $n$; the source presents it as a generalization rather than discussing its proof or resolution.