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

We present the main features of the mathematical theory generated by the κ-deformed exponential function exp κ(χ) = (√ 1 κ²χ² κχ)¹/κ, 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