On the Definition of Higher Gamma Functions

Abstract

We extent our definition of Euler Gamma function to higher Gamma functions, and we give a unified characterization of Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson q-Gamma function, and Nishizawa higher q-Gamma functions in the space of finite order meromorphic functions. The method extends to more general functional equations and unveils the multiplicative group structure of solutions that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. With the new definition, Barnes–Hurwitz zeta functions are no longer necessary in the definition of Barnes multiple Gamma functions. This simplifies the classical definition, without the 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 prove uniqueness of the solutions. Hence, this implies the identification of classical Barnes multiple Gamma functions as a subclass of our multiple Gamma functions.