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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(x)=(\sqrt{1+κ^2 x^2}+κx)^{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.