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

Author

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  • Niven, Robert K.
  • Suyari, Hiroki

Abstract

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>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≪∞.

Suggested Citation

  • Niven, Robert K. & Suyari, Hiroki, 2009. "The q-gamma and (q,q)-polygamma functions of Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(19), pages 4045-4060.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:19:p:4045-4060
    DOI: 10.1016/j.physa.2009.06.018
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