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We explore a number of functional properties of the $q$-gamma function and aclass of its quotients; including the $q$-beta function. We obtain formulas forall higher logarithmic derivatives of these quotients and give preciseconditions on their sign. We prove how these and other functional properties,such as the multiplication formula or the asymptotic expansion, together withthe fundamental functional equation of the $q$-gamma function uniquely definethose functions. We also study reciprocal "relatives" of the fundamental$q$-gamma functional equation, and prove uniqueness of solution results forthem. In addition, we also use a reflection formula of Askey to deriveexpressions relating the classical sine function and the number $\pi$ to the$q$-gamma function. Throughout we highlight the similarities and differencesbetween the cases $0<q<1$ and $q>1$.

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