  We study the three-dimensional lens partition function for $\mathcal N=2$
supersymmetric gauge dual theories on $S^3/\mathbb{Z}_r$ by using the gauge/YBE
correspondence. This correspondence relates supersymmetric gauge theories to
exactly solvable models of statistical mechanics. The equality of partition
functions for the three-dimensional supersymmetric dual theories can be written
as an integral identity for hyperbolic hypergeometric functions. We obtain such
an integral identity which can be written as the star-triangle relation for
Ising type integrable models and as the integral pentagon identity. The latter
represents the basic 2-3 Pachner move for triangulated 3-manifolds. A special
case of our integral identity can be used for proving orthogonality and
completeness relation of the Clebsch-Gordan coefficients for the self-dual
continuous series of $U_q(osp(1|2))$.
