  We generalize our previous new definition of Euler Gamma function to higher
Gamma functions. With this unified approach, we characterize Barnes higher
Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions,
Jackson $q$-Gamma function, and Nishizawa higher $q$-Gamma functions. This
approach extends to more general functional equations. This generalization
reveals the multiplicative group structure of solutions of the functional
equation that appears as a cocycle equation. We also generalize Barnes
hierarchy of higher Gamma function and multiple Gamma functions. In this new
approach, Barnes-Hurwitz zeta functions are no longer required for the
definition of Barnes multiple Gamma functions. This simplifies the classical
definition, without the necessary analytic preliminaries about the meromorphic
extension of Barnes-Hurwitz zeta functions, and defines a larger class of Gamma
functions. For some algebraic independence conditions on the parameters, we
have uniqueness of the solutions, which implies the coincidence of our multiple
Gamma functions with Barnes multiple Gamma functions.
