We study lens partitions functions for the three-dimensional N = 2 supersymmetric gauge theories on S 3 b /Z r . We consider equality as a new hyperbolic hypergeometric solution to the star-star relation via gauge/YBE correspondence. The correspondence allows the construction of integrable lattice spin models of statistical mechanics by the usage of integral identities. Additionally, we obtain new hyperbolic hypergeometric integral identities of gauge theories.