Functional definitions for $q$-analogues of eulerian functions and applications

Abstract

We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or the asymptotic expansion, together with the fundamental functional equation of the $q$-gamma function uniquely define those functions. We also study reciprocal "relatives" of the fundamental $q$-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number $\pi$ to the $q$-gamma function. Throughout we highlight the similarities and differences between the cases $0<q<1$ and $q>1$.