Bilinear relations for q-deformed conformal blocks

Let $q$ satisfy $|q|<1$, let $u$ be a parameter, and let $(u;q,q)_\infty$ denote the corresponding double $q$-Pochhammer product. The $q$-deformed conformal block $\mathcal{F}(u;q^{-1},q|Z)$ is the power-series function introduced above.

Bilinear relations. The $q$-deformed conformal blocks satisfy

$$\sum_{2n\in\mathbb{Z}} \frac{u^{2n}Z^{2n^2}}{\prod\limits_{\epsilon, \epsilon'=\pm1}(u^{\epsilon}q^{1+2\epsilon'n};q,q)_{\infty}}\mathcal{F}(uq^{-2n};q^{-1},q|q^{-1}Z)\mathcal{F}(uq^{2n};q^{-1},q|qZ) =(1-Z^{1/2})\sum_{2n\in\mathbb{Z}} \frac{Z^{2n^2}}{\prod\limits_{\epsilon, \epsilon'=\pm1}(u^{\epsilon}q^{1+2\epsilon'n};q,q)_{\infty}}\mathcal{F}(uq^{-2n};q^{-1},q|Z)\mathcal{F}(uq^{2n};q^{-1},q|Z).$$

These identities are part of the paper's bilinear structure for $q$-deformed conformal blocks; the supplied text gives no separate resolution status.