^{1, 2}

^{1}

^{3}

^{1}

^{2}

^{3}

From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs). 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. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.

The convenience of generalizing the logarithmic function has attracted the attention of researchers since long ago [

Here, our main objective is to show that the generalized stretched exponential function, written as probability density function (pdf), is suitable to generalize a wide range of one- and two-tail pdfs. This approach, which stresses the emergence of several probability distributions, is motivated by a mathematical curiosity. In Section

From the integration of nonsymmetrical hyperboles, we obtain a one-parameter generalization of the logarithmic function, which coincides with the one obtained in the context of nonextensive thermostatistics [

In the one-parameter generalization we address here, the

The usual natural logarithm (

The

This is a nonnegative function (

The derivative of the

Let us turn our attention to discrete random variables, the rank distribution in particular. We show that the rank distribution obtained by Naumis and Cocho [

To simultaneously fit the beginning, body and tail of experimental rank distributions of complex systems, Naumis and Cocho [

Finite size effects are described by factor

If one writes

In what follows, we first show that the Zipf-Mandelbrot function, which is a fingerprint of complex systems, can be written in terms of the

The envelope of a typical rank distribution of complex systems can be well described by the Zipf-Mandelbrot function [

We remark that (

A deformation in the argument of (

The usual stretched exponential function, also known as the Kohlrausch function [

The stretched exponential function can also be obtained as the solution of the following differential equation:

Considering the factor

If the considered independent variable

The normalization factor

The integral of (

For our purposes, it is more convenient to write

Behavior of (

Behavior of (

As

The cumulative function of (

is the hypergeometric function [

is the Pochhammer symbol.

The moments of

If

Particular values of

To generalize the error function, consider

An alternative way to obtain (

For

If the domain of the considered independent variable is not bounded, it is interesting to consider its absolute value

On one hand, due to its symmetry around

In the following, we retrieve the generalized Gaussian as a particular case of (

An interesting particular case of the two-tail generalized stretched exponential function (

Using that for

The characteristic function of (

For

The odd moments of (

We have shown that the

A. S. Martinez acknowledges the Brazilian agencies CNPq (303990/2007-4 and 476862/2007-8) for support. R. S. González also acknowledges CNPq (140420/2007-0) for support.