On Bailey pairs for $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on $$ {S}_b^3/{\mathbb{Z}}_r $$

We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions three-dimensional $\mathcal N=2$ supersymmetric gauge theories on $S_b^3/\mathbb{Z}_r$. The novel are constructed star-triangle relation, star-star relation and pentagon identity. first two them integrability conditions Ising-type integrable lattice models. last one corresponds to representation basic $2-3$ Pachner move triangulated 3-manifolds.