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) = (root 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.