  We prove that the K-theoretic Nekrasov instanton partition functions have a
positive radius of convergence in the instanton counting parameter and are
holomorphic functions of the Coulomb parameters in a suitable domain. We
discuss the implications for the AGT correspondence and the analyticity of the
norm of Gaiotto states for the deformed Virasoro algebra. The proof is based on
random matrix techniques and relies on an integral representation of the
partition function, due to Nekrasov, which we also prove.
