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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)

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# 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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

Kaniadakis, G. Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions. *Entropy* **2013**, *15*, 3983-4010.