Nonextensive Statistical Mechanics

Appendix A Useful Mathematical Formulae 333d 3 ∣ ∫F q [ f ](ξ) ∣∣∣ξ=0 ∞=−iq(2q − 1) dx x 3 [ f (x)] 3q−2dξ 3 −∞(q ≥ 1) (A.33)} ∫ ∞d (n) F q [ f ](ξ)dξ n∣ { n−1∣∣∣ξ=0 ∏= (i) n [1 + m(q − 1)]m=0dx x n [ f (x)] 1+n(q−1)−∞(q ≥ 1; n = 1, 2, 3...) (A.34)F q [af(ax)](ξ) = F q [ f ](ξ/a 2−q ) (a > 0; 1 ≤ q < 2) . (A.35)The generating function I (t) (t ∈ R) of a given distribution P N (N = 0, 1, 2, ...)is defined as follows:∞∑∞∑I (t) ≡ t N P N ( P N = 1) .N=0N=0(A.36)The negative binomial distribution is defined as follows:P N ( ¯N, k) ≡where(N + k − 1)!( ¯N/k) N (1) k( ¯N > 0, k > 0) , (A.37)N!(k − 1)! 1 + ¯N/k 1 + ¯N/k∞∑¯N = NP N ( ¯N, k) ,N=0(A.38)Its generating function is given by1k = [∑ ∞N=0 (N − ¯N) 2 P N ] − ¯N. (A.39)¯N 2withI (t) = e ¯N(t−1)q , (A.40)q ≡ 1 + 1 k .(A.41)The particular case q = 1 (i.e., k →∞) corresponds to the Poisson distribution