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

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