Generalized Exponential Function and some of its Applications to Complex Systems

TL;DR

This paper introduces a generalized exponential function derived from non-symmetrical hyperboles, demonstrating its utility in modeling complex systems, extending classical functions like gamma and factorial, and describing various probability distributions.

Contribution

It presents a novel generalized exponential function and applies it to extend classical functions and model complex system behaviors.

Findings

Generalized exponential function effectively models complex system functions.

Extended gamma and factorial functions using the new generalization.

Derived analytical forms for distributions like generalized error, Zipf-Mandelbrot, and generalized Gaussian.

Abstract

From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing complex systems can be conveniently written in terms of this generalization of the exponential function. The gamma function is then generalized and we generalize the factorial operation. Also a very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. One obtains, as particular cases, the generalized error function, the Zipf-Mandelbrot probability density function (pdf), the generalized gaussian and Laplace pdf. One can also obtain analytically their cumulative functions and moments.