Superintegrability for some (q, t)-deformed matrix models

Abstract We analyze the Macdonald’s ( q , t )-deformed hypergeometric functions with one and two set variables and present their constraints. We prove the uniqueness to the solutions of these constraints. We propose a concise method to prove the superintegrability relations for ( q , t )-deformed matrix models, where the constraints of hypergeometric functions play a crucial role. A conjectured superintegrability relation in the literature for the refined Chern–Simons model can be easily proved by our method. Moreover, we construct a general ( q , t )-deformed matrix model. We give the constraint conditions for parameters in the integral. The superintegrability relations for the ( q , t )-deformed integrals with allowed parameters are derived from the hypergeometric constraints.