Bilinear relation for generic q-deformed conformal blocks

Let $q_1,q_2,u,Z$ satisfy $|q_2|<1<|q_1|$. Define

$$\widehat{\mathcal{F}}_d(u,q_1,q_2|Z)=\sum_{2n\in\mathbb{Z}}\frac{u^{2dn}(q_1q_2)^{4dn^2}Z^{2n^2}}{(uq_1^{4n-2},u^{-1}q_1^{-4n-2})^{(1)}_\infty(uq_1^{-1}q_2^{4n+1},u^{-1}q_1^{-1}q_2^{-4n+1})^{(2)}_\infty}\mathcal{F}^{(1)}_n(q_1^{2d}Z)\mathcal{F}^{(2)}_n(q_2^{2d}Z),$$

where

$$\mathcal{F}^{(1)}_n(z)=\mathcal{F}(uq_1^{4n},q_1^2,q_1^{-1}q_2|z),\qquad \mathcal{F}^{(2)}_n(z)=\mathcal{F}(uq_2^{4n},q_1q_2^{-1},q_2^2|z).$$

Also define

$$\mathcal{F}_{\lozenge}(u,q_1,q_2|Z)=\sum_{\lambda_1,\lambda_2}Z^{\frac{|\lambda_1|+|\lambda_2|}{2}}\frac{1}{\prod_{i,j=1}^2N^{\mathrm{NS}}_{\lambda_i,\lambda_j}(q_1,q_2,u_i/u_j)}.$$

The source defines the factors $N^{\lozenge}_{\lambda,\mu}$ using the subsets of boxes selected by the stated parity condition.

Generic bilinear relation.

$$\mathcal{F}_{\lozenge}(u,q_1,q_2|Z)=\widehat{\mathcal{F}}_1(u,q_1,q_2|Z),\qquad \mathcal{F}_{\lozenge}(u,q_1,q_2|Z)=(1-q_2q_1Z^{1/2})\widehat{\mathcal{F}}_0(u,q_1,q_2|Z).$$

This extends the bilinear relations from $q_1=q^{-1}$ and $q_2=q$ to generic parameters in the indicated sector; the supplied text gives no separate resolution status.