Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions

We present the main features of the mathematical theory generated by the \kappa-deformed exponential function exp_{\kappa}(x)=(\sqrt{1+\kappa^2 x^2}+\kappa x)^{1/\kappa}, with 0<\kappa<1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The \kappa-mathematics has its roots in special relativity and furnishes the theoretical foundations of the \kappa-statistical mechanics predicting power law tailed statistical distributions which have been observed experimentally in many physical, natural and artificial systems. After introducing the \kappa-algebra we present the associated \kappa-differential and \kappa-integral calculus. Then we obtain the corresponding \kappa-exponential and \kappa-logarithm functions and give the \kappa-version of the main functions of the ordinary mathematics.

In this contribution we present the main features of the mathematical theory generated by the function exp κ (x). The κ-mathematics, developed in the last twelve years, turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical theory predicting power law tailed statistical distributions which have been observed experimentally in many physical, natural and artificial systems.
The paper is organized as follows: After introducing the κ-algebra we present the associated κ-differential and κ-integral calculus. Then we obtain the corresponding κ-generalized exponential and logarithmic functions and give the κ-version of the main functions of the ordinary mathematics.
Proof. From the definition of κ ⊕ the following properties follow 1) associativity: Remarks. The κ-sum is a one parameter continuous deformation of the ordinary sum which recovers in the classical limit κ → 0, i.e. x 0 ⊕ y = x + y. The κ-sum (2.1) is the additivity law of the dimensionless relativistic momenta of special relativity while the real parameter −1 < κ < 1 is the reciprocal of the dimensionless light speed [3,9]. The Theorem II.2. Let be x, y ∈ R and −1 < κ < 1. The composition law κ ⊗ defined through is a generalized product, called κ-product and the algebraic structure (R, κ ⊗) forms an abelian group.
Proof. From the definition of κ ⊗ the following properties follow 1) associativity: (x κ ⊗ y) 2) neutral element: is defined through x κ ⊗ I = I κ ⊗ x = x and is given by Remarks. The κ-product reduces to the ordinary product as κ → 0, i.e.
Theorem II.3. Let be x, y ∈ R and −1 < κ < 1. The κ-sum κ ⊕ defined in (2.1), and the κ-product The κ-differential of x, indicated by d κ x, is defined through and results to be In order to better understand the origin of the expression of d κ x we recall that the variable x is a dimensionless momentum. Then the quantity γ(x) = √ 1 + κ 2 x 2 is the Lorentz factor of relativistic physics, in the momentum representation. So, we can write Moreover it holds We define the κ-derivative of the function f (x) through We observe that df (x)/d κ x, which reduces to df (x)/dx as the deformation parameter κ → 0, can be written in the form From the latter relatioship follows that the κ-derivative obeys the Leibniz's rules of the ordinary derivative. After introducing the γ(x) Lorentz factor the κ-derivative can be written also in the form: We define the κ-integral as the inverse operator of the κ-derivative through 8) and note that it is governed by the same rules of the ordinary integral, which recovers when κ → 0.

D. Connections with Physics
We indicate with p = |p| and x = p/mv * the moduli of the particle momentum in dimensional and dimensionless form respectively and define κ = v * /c. The classical relationship linking x with the dimensionless kinetic energy W = x 2 /2 follows from the kinetic energy theorem, which in differential form reads The latter equation after replacing the ordinary derivative by the derivative d/d κ x, i.e.
transforms into the corresponding relativistic equation. This differential equation with the condition W(x = 0) = 0 admits as unique solution W = √ 1 + κ 2 x 2 − 1 /κ 2 defining the relativistic kinetic energy. Let us consider the four-dimensional Lorentz invariant integral being p µ = (p 0 , p) = m 2 c 2 + p 2 , p , θ(.) the Heaviside step function and δ(.) the Dirac delta function. It is trivial to verify that the latter integral transforms into the one-dimension integral being f (x) = 4πx 2 F (x). Then the κ-integral is essentially the Lorentz invariant integral of special relativity.

A. Definition
We recall that the ordinary exponential f (x) = exp(x) emerges as solution both of the functional equation f (x+y) = f (x)f (y) and of the differential equation (d/dx)f (x) = f (x). The question to determine the solution of the generalized equations reducing in the κ → 0 limit to the ordinary exponential, naturally arises. This solution is unique and represents a one-parameter generalization of the ordinary exponential.

Solution of Eq. (4.1): We write this equation explicitly
which after performing the change of variables f (x) = exp(g(κx)), z 1 = κx, z 2 = κy transforms as and admits as solution

Solution of Eq. (4.2):
After recalling that d κ x = d{x} with {x} = κ −1 arcsinh κx, Eq. (4.6) can be written in the form The solution of the latter equation with the condition exp κ (0) = 1 follows immediately After taking into account that arcsinh x = ln( which will used in the following. We remark that exp κ (x) given by Eq. (4.9), is solution both of the Eqs. (4.1) and (4.2) and therefore represents a generalization of the ordinary exponential.

B. Basic Properties
From the definition (4.9) of exp κ (x) follows that Like the ordinary exponential, exp κ (x) has the properties (4.17) In Fig.1 is plotted the function exp κ (x) defined in Eq.(4.9) for three different values of the parameter of κ. The continuous curve corresponding to κ = 0 is the ordinary exponential function exp(x). The property (4.17) emerges as particular case of the more general one with r ∈ R, which in the limit κ → 0 reproduces one well known property of the ordinary exponential. We remark the following convexity property holding when κ 2 < 1.
Undoubtedly one of the more interesting properties of exp κ (x) is its power law asymptotic behavior Let us consider the incomplete Mellin transform of the exp κ (−t) After performing the change of integration variable y = √ 1 + κ 2 t 2 − |κ|t 2 , and after taking into account that t = 1 2|κ| y −1/2 − y 1/2 and exp κ (−t) = y 1/2|κ| , the function M κ (r, x) can be written in the form When r is an integer greater that zero, M κ (r, x) can be calculated analytically. For instance it results In general the function M κ (r, x) can be written as with After recalling the definition of the Beta incomplete function B X (s, r) = X 0 y s−1 (1 − y) r−1 dy, the integral I X (r) becomes The function I 1 (r) can be expressed in term of the Beta functions B (s, r) = 1 0 y s−1 (1 − y) r−1 dy, and then in terms of Gamma functions, obtaining The Mellin transform of exp κ (−t), namely can be calculated from Eq. (4.31) by posing x = ∞. The explicit expression of M κ (r) holding for 0 < r < 1/|κ| is given by From the latter relationship one can verify easily the property (4.34)

D. Taylor expansion
The Taylor expansion of exp κ (x) given in [3] can be written also in the following form where the symbol n! κ , representing the κ-generalization of the ordinary factorial n!, recovered for κ = 0, is given by  The polynomials ξ n (κ), for n > 1, when n is odd, are of degree n − 1, with respect the variable κ, while when n is even the degree of ξ n (κ) is n − 2. The degree of ξ n (κ) is always an even number and it results The polynomials ξ n (κ) can be generated by the following simple recursive formula (4.42) The first nine polynomials reads as (4.49) After noting that for a given value of κ the maximum natural number N satisfying the condition N < 2 + 1/|κ| is defined univocally, we can verify easily that for n = 0, 1, 2, ..., N it results ξ n (κ) > 0 and then n! κ > 0. For n > N the sign of ξ n (κ) and then of n! κ alternates with periodicity − − + + − − + + ... From Eqs. (4.36) and (4.42) follows the recursive formula (n + 2)! κ = (n + 1)(n + 2) 1 − n 2 κ 2 n! κ . (4.50) By direct comparison of Eq. (4.34) and (4.50) we obtain the relationship It is remarkable that the first three terms in the Taylor expansion of exp κ (x) are the same of the ordinary exponential The Γ κ (n) with n integer is defined through and represents a generalization of the Euler Γ(n) function. In particular we have Γ κ (1) = Γ κ (2) = 1 and Γ κ (3) = 2. This definition and the relationship (4.51) suggests the following one parameter generalization of the Euler Γ(x) function i.e. Γ κ (x), given by The explicit expression of Γ κ (x) in terms of the ordinary Γ(x) is given by and can be used as definition of Γ κ (x) when x is a complex variable. Clearly in the κ → 0 limit it results Γ 0 (x) = Γ(x). An expression of Γ κ (x) in terms of the Beta function is the following (4.57) The Taylor expansion of exp κ (x) can be written also in the form x n Γ κ (n + 1) ; κ 2 x 2 < 1 . In Fig.2 and Fig.3 is plotted the function Γ κ (x) defined in Eq.(4.55) in the ranges −4 < x < 4 and 9 < x < 12 respectively, for κ = 0 and κ = 0.15. The continuous curve corresponding to κ = 0 is the ordinary Gamma function Γ(x).
The incomplete γ κ (r, x) and Γ κ (r, x) are defined as and hold the following relationships (4.63)

F. Expansion in ordinary exponentials
Starting from the expression (4.5) exp κ (x) and the Taylor expansion of the function arcsinh(x) we obtain with c n = (−1) n (2n)! (2n+1) 2 2n (n!) 2 . (4.65) Exploiting this relationship, we can write exp κ (x) as an infinite product of ordinary exponentials On the other hand exp κ (x) can be viewed as a continuous linear combination of an infinity of standard exponentials. Namely for Re s ≥ 0, it holds the following Laplace transform being J ν (x) the Bessel function.

G. κ-Laplace transform
The following κ-Laplace transform emerges naturally as a generalization of the ordinary Laplace transform. The inverse κ-Laplace transform is given by In ref. [34] the mathematical properties of the κ-Laplace transform have been investigated systematically. In table I are reported the main properties of the κ-Laplace transform which in the κ → 0 limit reduce to the corresponding ordinary Laplace transform properties.
Furthermore it holds the initial value theorem  The κ-convolution of two functions f and has the following properties It holds the following κ-convolution theorem In table II are reported the κ-Laplace transforms for the delta function, for the unit function and for the power function. We note that κ-Laplace transform of the power function f (t) = t ν−1 involves the κ-generalized Gamma function. All the κ-Laplace transforms in the κ → 0 limit reduce to the corresponding ordinary Laplace transforms.

A. Definition and basic properties
The function ln κ (x) is defined as the inverse function of exp κ (x), namely ln κ (exp κ x) = exp κ (ln κ x) = x , (5.1) and is given by or more properly It results that The function ln κ (x), just as the ordinary logarithm, has the properties Furthermore ln κ (x) has the two properties ln κ (x r ) = r ln rκ (x) , (5.13) ln κ (x y) = ln κ (x) ⊕ κ ln κ (y) , (5.14) with r ∈ R. Note that the property (5.12) follows as particular case of the property (5.14).
We remark the following concavity properties A very interesting property of this function is its power law asymptotic behavior one can verify that the latter relationship can be generalized easily in order to obtain ln κ (x), by replacing the integrand function y 0 (t) = t −1 by the new function y κ (t) = t −1−κ , namely (5.20)

B. Taylor expansion
The Taylor expansion of ln κ (1 + x) converges if −1 < x ≤ 1, and assumes the form with b 1 (κ) = 1, while for n > 1, b n (κ) is given by It results b n (0) = 1 and b n (−κ) = b n (κ). The first terms of the expansion are The following integral is useful Starting from the definition of the generalized Euler gamma function, i.e. Γ κ (x) given in the previous section, we can write it also in the following alternative but equivalent form An expression of Γ κ (x), where the parameter κ enters exclusively through the function ln κ (.), follows easily From the latter relationships follows that n! κ is given by The logarithm y(x) = ln(x) is the only existing function, unless a multiplicative constant, which results to be solution of the function equation y(x 1 x 2 ) = y(x 1 ) + y(x 2 ). Let us consider now the generalization of this equation, obtained by substituting the ordinary sum by the generalized sum We proceed by solving this equation, which assumes the explicit form After performing the substitution y(x) = κ −1 sinh κg(x) we obtain that the auxiliary function g(x) obeys the equation g(x 1 x 2 ) = g(x 1 ) + g(x 2 ), and then is given by g(x) = A ln x. The unknown function y(x) becomes y(x) = κ −1 sinh(κ ln x) where we have set A = 1 in order to recover, in the limit κ → 0, the classical solution y(x) = ln(x). Then we can conclude that the solution of Eq. (5.28) is given by lnκ(x) as solution of a differential-functional equation The following first order differential-functional equation emerges in statistical mechanics within the context of the maximum entropy principle The latter problem admits two solutions [3,9]. The first is given by f (x) = ln(x) and γ = 1, ǫ = e. The second solution is given by and The constant γ is the Lorentz factor corresponding to the reference velocity v * while the constant ǫ = exp κ (γ), represents the κ-generalization of the Napier number e.

F. The Entropy
A physically meaningful link between the functions ln κ (x) and exp κ (x) is given by a variational principle. It holds the following theorem: Theorem: Let be g(x) an arbitrary real function and y(x) a probability distribution function of the variable x ∈ A. The solution of the variational equation is unique and is given by being the constants γ and ǫ defined by Eqs. (5.36) and (5.37) respectively. The proof of the theorem is trivial and employs Eqs. (5.31). This theorem permits us to interpret the functional which can be written also in the form as the entropy associated to the function exp κ (x). It is remarkable that in the κ → 0 limit, as ln κ (y) and exp κ (x) approach ln(y) and exp(x) respectively, the new entropy reduces to the old Boltzmann-Shannon entropy. It is shown that the entropy S κ has the standard properties of Boltzmann-Shannon entropy: is thermodynamically stable, is Lesche stable, obeys the Khinchin axioms of continuity, maximality, expandability and generalized additivity.

A. κ-Hyperbolic Trigonometry
The κ-hyperbolic trigonometry can be introduced by defining the κ-hyperbolic sine and κ-hyperbolic cosine starting from the κ-Euler formula exp κ (±x) = cosh κ (x) ± sinh κ (x) .  The κ-hyperbolic tangent and cotangent functions are defined through Holding the relationships sinh κ (x) = sinh ({x}) , (6.6) cosh κ (x) = cosh ({x}) , (6.7) tanh κ (x) = tanh ({x}) , (6.8) coth κ (x) = coth ({x}) , (6.9) it is straightforward to verify that κ-hyperbolic trigonometry preserves the same structure of the ordinary hyperbolic trigonometry which recovers as special case in the limit κ → 0. For instance from the κ-Euler formula and from exp κ (−x) exp κ (x) = 1 the fundamental formula of the κ-hyperbolic trigonometry follows All the formulas of the ordinary hyperbolic trigonometry still hold, after proper generalization. Taking into account that {x κ ⊕ y} = {x} + {y}, it is easy to verify that the generalization of a given formula can be obtained starting from the corresponding ordinary formula, and then by making in the arguments of the hyperbolic trigonometric functions the substitutions x + y → x κ ⊕ y and x − y → x κ ⊖ y. For instance it results , (6.13) and so on.
κ ⊗ x is required, so that, for instance it holds the formula and so on. The κ-De Moivre formula involving hyperbolic trigonometric functions having arguments of the type rx, with r ∈ R, assumes the form Also the formulas involving the derivatives of the hyperbolic trigonometric function still hold, after properly generalized. For instance we have and so on. The κ-inverse hyperbolic functions can be introduced starting from the corresponding ordinary functions. It is trivial to verify that κ-inverse hyperbolic functions are related to the κ-logarithm by the usual formulas of the ordinary mathematics. For instance we have and consequently hold arcsinh κ (x) = arccosh κ 1 + x 2 , (6.22) involving the function arcsinh κ (x), follows from Eq. (6.18). Analogous formulas involving arccosh κ (x), arctanh κ (x) or arccoth κ (x) do not hold instead.