Algebraic-solution relation for q-deformed conformal blocks

Let $\mathcal{T}(u,s;q|Z)$ be any $u$-inverse invariant tau function, and let $C(u;q|Z)$ be its coefficient function. Here $u$-inverse invariance means invariance under $u\mapsto u^{-1}$, and $P_n(q)$ is defined by

$$P_n(q)=\frac{\prod\limits_{\epsilon=\pm1}(q^{\frac12\epsilon}q;q,q)_\infty}{\prod\limits_{\epsilon=\pm1}((q^{2n+1/2})^\epsilon q;q,q)_\infty}.$$

For $n>0$, set $k=2n$, and for $n<0$, set $k=-2n-1$.

Algebraic-solution relation. For $s=\pm1$,

$$\mathcal{T}(q^{1/2},\pm1;q|Z)=\frac{C(q^{1/2};q|Z)}{(q^{3/2};q,q)_\infty(q^{1/2};q,q)_\infty}(\mp Z^{1/2}q^{1/2};q^{1/2},q^{1/2})_\infty.$$

Equivalently,

$$(\mp Z^{1/2}q^{1/2};q^{1/2},q^{1/2})_\infty=\sum_{n\in\mathbb{Z}}(\mp1)^nZ^{n^2+n/2}P_n(q)\mathcal{F}(q^{2n+1/2},q,q|Z),$$

where

$$P_n(q)=\prod_{j=0}^{k-1}\frac{1}{\left((1-q^{j+1/2})(1-q^{-j-1/2})\right)^{k-j}}.$$

This relation gives the special algebraic, Bäcklund-invariant solutions associated with $u=q^{1/2}$ and $s=\pm1$; the supplied text gives no separate resolution status.