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# Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received: 29 July 2013 / Revised: 17 September 2013 / Accepted: 17 September 2013 / Published: 25 September 2013

(This article belongs to the collection Advances in Applied Statistical Mechanics)

# Abstract

We present the main features of the mathematical theory generated by the

*κ*-deformed exponential function ${\mathrm{exp}}_{k}\left(x\right)\text{}=\text{}{(\sqrt{1\text{}+\text{}{k}^{2}{x}^{2}}\text{}+\text{}kx)}^{\frac{1}{k}}$, with 0

*≤*

*κ <*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

*κ*-mathematics has its roots in special relativity and furnishes the theoretical foundations of the

*κ*-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the

*κ*-algebra, we present the associated

*κ*-differential and

*κ*-integral calculus. Then, we obtain the corresponding

*κ*-exponential and

*κ*-logarithm functions and give the

*κ*-version of the main functions of the ordinary mathematics.

*Keywords:*

*κ*-statistical mechanics;

*κ*-mathematics;

*κ*-exponential;

*κ*-logarithm; power-law tailed statistical distributions

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*