We study the three-dimensional lens partition function for <math><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></math> supersymmetric gauge dual theories on <math><msup><mi>S</mi><mn>3</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>r</mi></msub></math> by using the gauge/Yang-Baxter equation 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 <math><msub><mi>U</mi><mi>q</mi></msub><mo>(</mo><mi>o</mi><mi>s</mi><mi>p</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>2</mn><mo>)</mo><mo>)</mo></math>.