An axiomatic definition is given for the q-gamma function Γ q ( x ) , q ∈ R , q > 0 , x ∈ R of Tsallis (non-extensive) statistical physics, the continuous analogue of the q-factorial of Suyari [H. Suyari, Physica A 368 (1) (2006) 63], and the q-analogue of the gamma function Γ ( x ) of Euler and Gauss. A working definition in closed form, based on the Hurwitz and Riemann zeta functions (including their analytic continuations), is shown to satisfy this definition. Several relations involving the q-gamma and other functions are obtained. The (q,q)-polygamma functions ψ q , q ( m ) ( x ) , m ∈ N , defined by successive derivatives of ln q Γ q ( x ) , where ln q a = ( 1 − q ) − 1 ( a 1 − q − 1 ) , a > 0 is the q-logarithmic function, are also reported. The new functions are used to calculate the inferred probabilities and multipliers for Tsallis systems with finite numbers of particles N ≪ ∞ .