Flipping relation as a reduced star-star relation

In this paper, we consider the lens hyperbolic gamma solution to the star-star relation and the flipping relation from three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3\mathrm{/}{\mathbb{Z}}_r$. We explore that a certain limit of the star-star relation yields the latter symmetry transformation, which exchanges the edge interactions of two outer spins with a centrally sited spin. Furthermore, we obtain more solutions to the flipping relation in terms of the hyperbolic gamma, basic hypergeometric, and the Euler gamma functions.