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Useful Mathematical Formulae

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Useful Mathematical Formulae Appendix A Useful Mathematical Formulae x1−q − 1 ln x ≡ (x > 0, q ∈ R)(A.1) q 1 − q 1−q lnq x = x ln2−q x (x > 0; ∀q)(A.2) lnq (1/x) + ln2−q x = 0(x > 0; ∀q)(A.3) q q lnq x + ln(1/q)(1/x ) = 0(x > 0; ∀q)(A.4) ⎧ < < − / − , ⎪ 0ifq 1 and x 1 (1 q) ⎪ ⎪ ⎪ 1 ⎨⎪ + − 1−q < ≥− / − , 1 [1 (1 q) x] if q 1 and x 1 (1 q) x 1−q e ≡ [1 + (1 − q) x]+ ≡ q ⎪ ⎪ ex if q = 1(∀x) , ⎪ ⎪ ⎩ 1 [1 + (1 − q) x] 1−q if q > 1 and x < 1/(q − 1) . (A.5) x −x = ∀ eq e2−q 1(q)(A.6) x q −qx = ∀ (eq ) e(1/q) 1(q)(A.7) x+y+(1−q)xy = x y ∀ eq eq eq ( q)(A.8) 329 330 Appendix A Useful Mathematical Formulae x ⊕q y ≡ x + y + (1 − q) xy (A.9) For x ≥ 0 and y ≥ 0: ⎧ −q −q ⎪ 0ifq < 1andx 1 + y1 < 1 , ⎪ ⎪ ⎪ 1 ⎪ 1−q 1−q 1−q 1−q 1−q 1 ⎨ [x + y − 1] if q < 1andx + y ≥ 1 , 1−q 1−q 1−q x ⊗ y ≡ [x + y − 1]+ ≡ q ⎪ ⎪ xy if q = 1 ∀(x, y) , ⎪ ⎪ ⎩ − − 1 − − [x 1 q + y1 q − 1] 1−q if q > 1andx 1 q + y1 q > 1 . (A.10) 1 − x ⊗q y = [1 + (1 − q)(lnq x + lnq y)] 1 q (A.11) ⊕ x q y = x y ∀ eq eq eq ( q) (A.12) x+y = x ⊗ y ∀ eq eq q eq ( q) (A.13) d ln x 1 q = (x > 0; ∀q) (A.14) dx xq dex q = (ex )q (∀q) (A.15) dx q x q = qx ∀ (eq ) e2−(1/q) ( q) (A.16) x a = ax ∀ (eq ) e1−(1−q)/a ( q) (A.17) / − 2 − x 1/(q−1) 1 − b (q 1) a b b a− x x e = x q−1 e (b > 0; q > 1) (A.18) q q − 1 q Appendix A Useful Mathematical Formulae 331 1 1 3 1 2 1 ex = ex [1 − (1 − q)x2 + (1 − q)2x3(1 + x) − (1 − q)3x4(1 + x + x2) q 2 3 8 4 3 12 1 65 5 5 + (1 − q)4x5(1 + x + x2 + x3) 5 72 24 384 1 11 17 1 1 − (1 − q)5x6(1 + x + x2 + x3 + x4) + ...](q → 1; ∀x) 6 10 48 24 640 (A.19) 1 1 1 ln x = ln x [1 + (1 − q)lnx + (1 − q)2 ln2 x + (1 − q)3 ln3 x q 2 6 24 1 1 + (1 − q)4 ln4 x + (1 − q)5 ln5 x + ...](q → 1; x > 0) 120 720 (A.20) x ⊗q y = xy 1 − (1 − q)(ln x)(ln y) 1 + (1 − q)2 [(ln2 x)(ln y) + (ln x)(ln2 y) + (ln2 x)(ln2 y)] 2 1 − (1 − q)3 [2(ln3 x)(ln y) + 9(ln2 x)(ln2 y) + 2(ln x)(ln3 y) 12 +6(ln3 x)(ln2 y) + 6(ln2 x)(ln3 y) + 2(ln3 x)(ln3 y)] 1 + (1 − q)4 [(ln4 x)(ln y) + 14(ln3 x)(ln2 y) 24 +14(ln2 x)(ln3 y) + (ln x)(ln4 y) + 4 2 + 3 3 + 2 4 7(ln x)(ln y) 24(ln x)(ln y) 7(ln x)(ln y) +6(ln4 x)(ln3 y) + 6(ln3 x)(ln4 y) + (ln4 x)(ln4 y)] + ... (A.21) 1 1 1 ex = 1 + x + x2q + x3q(2q − 1) + x4q(2q − 1)(3q − 2) q 2 6 24 1 + x5q(2q − 1)(3q − 2)(4q − 3) + ... (x → 0; ∀q) (A.22) 120 1 1 1 ln (1 + x) = x − x2q + x3q(1 + q) − x4q(1 + q)(2 + q) q 2 6 24 1 + x5q(1 + q)(2 + q)(3 + q) + ... (x → 0; ∀q) (A.23) 120 x 1 b/(q−1)2 − q− 1 − a b b 1 a− x x e = x q−1 e (q > 1; b > 0) (A.24) q q − 1 q 332 Appendix A Useful Mathematical Formulae lnq x 2 lnq,q x ≡ lnq e (x > 0, (q, q ) ∈ R ) (A.25) 2 lnq,q (x ⊗q y) = lnq,q x ⊕q lnq,q y (x > 0, (q, q ) ∈ R ) (A.26) ∞ −β 1 1 2−q − α −α z = ααq−1 β(q−1) z eq 1 d e e (A.27) ⌫ 1 − q−1 [β(q − 1)] q 1 0 (α>0; β>0; 1 < q < 2) The following relations are useful for the Fourier transform of q-Gaussians (with β>0): ∞ eixp F (p) ≡ dx q + − β 2 1/(q−1) −∞ [1 (q 1) x ] ∞ cos (xp) = 2 dx 2 1/(q−1) 0 [1 + (q − 1) β x ] ⎧ √ 3−q ⎪ π 2−q 2 β(1−q) 2(1−q) p ⎪ ⌫ − √ < , ⎪ − β ( − ) J 3 q β − if q 1 ⎪ (1 q) 1 q p 1−q (1 q) ⎪ ⎪ ⎨ 2 π − p = e 4β if q = 1 , ⎪ β (A.28) ⎪ ⎪ ⎪ 3−q ⎪ π |p| 2(q−1) |p| ⎩ 2 √ − √ < < , β − β − K 3 q β − if 1 q 3 ⌫ 1 (q 1) 2 (q 1) 2(q−1) (q 1) q−1 where Jν (z) and Kν (z) are, respectively, the Bessel and the modified Bessel func- tions. For the three successive regions of q we have respectively used formulae 3.387-2 (page 346), 3.323-2 (page 333) and 8.432-5 (page 905) of [228] (see also [868]). For the q < 1 result we have taken into account the fact that the q-Gaussian |x| > √ 1 identically vanishes for β(1−q) . ∞ ∞ ξ ≡ iξ x ⊗ = iξ x[ f (x)]q−1 ≥ Fq [ f ]( ) dx eq q f (x) dx eq f (x)(q 1) (A.29) −∞ −∞ ∞ Fq [ f ](0) = dx f(x)(q ≥ 1) (A.30) −∞ ξ ∞ dFq [ f ]( ) q = i dx x [ f (x)] (q ≥ 1) (A.31) dξ ξ= −∞ 0 d2 F [ f ](ξ) ∞ q =−q dx x2 [ f (x)]2q−1 (q ≥ 1) (A.32) ξ 2 d ξ=0 −∞ Appendix A Useful Mathematical Formulae 333 d3 F [ f ](ξ) ∞ q =−iq(2q − 1) dx x3 [ f (x)]3q−2 (q ≥ 1) (A.33) ξ 3 d ξ=0 −∞ − d(n) F [ f ](ξ) n1 ∞ q = i n + m q − dx xn f x 1+n(q−1) n ( ) [1 ( 1)] [ ( )] dξ ξ= −∞ 0 m=0 (q ≥ 1; n = 1, 2, 3...) (A.34) 2−q Fq [af(ax)](ξ) = Fq [ f ](ξ/a )(a > 0; 1 ≤ q < 2) . (A.35) The generating function I (t)(t ∈ R) of a given distribution PN (N = 0, 1, 2, ...) is defined as follows: ∞ ∞ N I (t) ≡ t PN ( PN = 1) . (A.36) N=0 N=0 The negative binomial distribution is defined as follows: (N + k − 1)! N¯ /k N 1 k PN (N¯ , k) ≡ (N¯ > 0, k > 0) , (A.37) N!(k − 1)! 1 + N¯ /k 1 + N¯ /k where ∞ N¯ = NPN (N¯ , k) , (A.38) N=0 ∞ 1 [ (N − N¯ )2 P ] − N¯ = N=0 N . (A.39) k N¯ 2 Its generating function is given by = N¯ (t−1) , I (t) eq (A.40) with 1 q ≡ 1 + . (A.41) k The particular case q = 1 (i.e., k →∞) corresponds to the Poisson distribution 334 Appendix A Useful Mathematical Formulae N¯ N P (N¯ ) = e−N¯ , (A.42) N N! which satisfies the property that the width equals the mean value, i.e., ∞ 2 (N − N¯ ) PN = N¯ . (A.43) N=0 Appendix B Escort Distributions and q-Expectation Values B.1 First Example In order to illustrate the practical utility and peculiar properties of escort distribu- tions and their associated q-expectation values, we introduce and analyze here a pedagogical example [884].1 Let us assume that we have a set of empirical distributions { fn(x)} (n = 1, 2, 3, ...) defined as follows: A f (x) = n (λ>0; α ≥ 0) , (B.1) n (1 + λx)α if 0 ≤ x ≤ n, and zero otherwise. Normalization of fn(x) immediately yields λ(α − 1) A = . (B.2) n 1 − (1 + λ n)1−α In order to have finite values for An, ∀n, including n →∞(i.e., 0 < A∞ < ∞), α>1 is needed. Consequently A∞ = λ(α − 1) . (B.3) By identifying 1 α = , (B.4) q − 1 λ = β(q − 1) , (B.5) 1 The present illustration has greatly benefited from lengthy discussions with S. Abe, who launched [885] interesting questions regarding q-expectation values, and with E.M.F. Curado. 335 336 Appendix B Escort Distributions and q-Expectation Values Fig. B.1 The distributions fn (x)forn = 1, 2, 3, ∞ (from top to bottom) for (λ, α) = (2, 3/2) (from [884]). Equation (B.1) can be rewritten as = −β x β> ≥ . fn(x) An eq ( 0; q 1) (B.6) The variable x ≥ 0 could be a physical quantity, say earthquake intensity, mea- sured along small intervals, say 10−6, so small that sums can be replaced by integrals within an excellent approximation. The empiric distribution fn(x) could correspond to different seismic regions, say region 1 (for n = 1), region 2 (for n = 2), and so on. See Fig. B.1. Suppose we want to characterize the distribution fn(x) through its mean value. A straightforward calculation yields n 1 − (1 + λn)α + λn[α + (α − 1)λn] x (n) ≡ dx xf (x) = . (B.7) n α−1 0 (α − 2)λ(1 + λn)[1 − (1 + λn) ] This quantity is finite for all n (including n →∞)forα>2, but x (∞) diverges for 1 <α≤ 2.
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