q^{\frac{g-\ell+1}2} & \nu^{2}/q & \mu \nu q^{g-\ell} & \nu & q^{-L}
            \\
            & \nu q^{\frac{g-\ell-1}2} &  - \nu q^{\frac{g-\ell-1}2} & q^{g-\ell+1} & \nu/\mu  & \nu q^{g-\ell} & \nu^{2}q^{L+g-\ell} & \nu^{2}q^{g}
        \end{array};
        q, \frac{q^{L+g+1}}{\mu} \right].
    \end{gathered}$$

When $\ell\ge g$, we must have $L\ge \ell-g$. Let us rewrite in another form: $$\begin{gathered}
        \label{P_ell_g_rewrite_second_case}
                \mathbf{P}_{\ell,g}(L)
        =
        \mu^{L-\ell+g}
        \frac
        {(q^{\ell}; q^{-1})_{g} (\nu \mu^{-1} q^{\ell-g};q)_{L-\ell+g} (\nu; q)_{\ell-g}(\nu q^{g}; q)_{L-\ell+g}}
        {(\mu \nu; q)_{g} (q; q)_{L-\ell+g}(\nu^{2} q^{g}; q)_{L}}
        \\ \times
        {_{4}}\phi_{3}
        \left[\begin{array}{cccc}
                q^{-g} & \mu^{-1} \nu^{-1} q^{\ell-g+1} &  \nu
                q^{\ell-g} &  q^{-L+\ell-g}
                \\
                & \nu^{-1}q^{-L+\ell-2g+1} &  \nu \mu^{-1}
                q^{\ell-g} & q^{\ell-g+1}
            \end{array}; q, q
        \right]
        \frac{(\mu;
        q)_{\infty}(\nu^{2} q^{g}; q)_{\infty}}{(\nu; q)_{\infty}(\nu \mu q^{g};
        q)_{\infty}}.
    \end{gathered}$$ Again, the equality between and is termwise up to an index shift by $\ell-g$. Using Watson’s transformation formula , we get $$\begin{aligned}
                \nonumber&
        \mathbf{P}_{\ell,g}(L)
        =
        \frac
        {(\mu; q)_{\infty}(\nu^{2} q^{g}; q)_{\infty}}
        {(\nu; q)_{\infty}(\nu \mu q^{g}; q)_{\infty}}
        \\
        \label{P_ell_g_rewrite_second_case_87}
        &\hspace{10pt}\times
        \mu^{L-\ell+g}\,
        \frac{(q^{\ell}; q^{-1})_{g} (\nu \mu^{-1} q^{\ell-g};q)_{L-\ell+g} (\nu; q)_{\ell-g}
        (\nu q^{g};q)_{L-\ell+g}(\nu^{-2}q^{-L-g+1}; q)_{g} (\nu^{-1}q^{-g+1}; q)_{g}}
        {(\mu \nu;q)_{g} (q; q)_{L-\ell+g}(\nu^{2} q^{g}; q)_{L} (\nu^{-1} q^{-L+\ell-2g+1};q)_{g}
        (\nu^{-2}q^{-\ell+1}; q)_{g}}
        \\ &\hspace{10pt}
        \nonumber
        \times {_{8}}\phi_{7}
        \left[\begin{array}{cccccccc}
                q^{-g} & \nu^{2} q^{\ell-g-1} & \nu
                q^{\frac{\ell-g+1}2} & -\nu q^{\frac{\ell-g+1}2} & \nu^{2}/q & \mu \nu & \nu
                q^{\ell-g} & q^{-L+\ell-g}
                \\
                & \nu q^{\frac{\ell-g-1}2} &  - 