The q-gamma and (q,q)-polygamma functions of Tsallis statistics

An axiomatic definition is given for the q  -gamma function Γq(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 ψq,q(m)(x),m∈N, defined by successive derivatives of lnqΓq(x), where lnqa=(1−q)−1(a1−q−1),a>0lnqa=(1−q)−1(a1−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≪∞N≪∞.