# Generalized Probability Functions

Authors: Alexandre Souto Martinez; Rodrigo Silva González; César Augusto Sangaletti Terçariol
Journal: Advances in Mathematical Physics
 (2009)

---

## Abstract

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.


---

## Body

## 1. Introduction

The convenience of generalizing the logarithmic function has attracted the attention of researchers since long ago [1] and particularly in the last years [2–7]. In Physics, several one-parameter generalizations of the logarithmic function have been proposed in different contexts such as nonextensive statistical mechanics [8–13], relativistic statistical mechanics [14, 15], and quantum group theory [16]. Also, more sophisticated such as two-parameter [17] and three-parameter [18] generalizations have been proposed, each one including previous situations as particular cases. Examples of the convenience of these generalizations have been seen in different fields, for instance, psychophysics [19], neuroeconomics [20, 21], econophysics [22, 23], complex networks [24, 25], population dynamics [26, 27], and so forth.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 Section2, we show that from the integration of nonsymmetrical hyperboles, one obtains a one-parameter generalization of the logarithmic function, which we call q̃-logarithm. This generalization coincides with the one obtained in the context of nonextensive thermostatistics [9, 10]. Inverting the q̃-logarithm, one obtains the generalized exponential (q̃-exponential) function. Some properties of these generalized functions are presented. In Section 3, the very reliable rank distribution obtained by Naumis and Cocho [28] is conveniently described by the q̃-exponential function, which permits us to detect the effect of finite sample size in the description. In Section 4, we first show that the Zipf-Mandelbrot function, which is a fingerprint of complex systems, can be conveniently written in terms of the q̃-exponential. Raising the q̃-exponential argument to a given power, one obtains a function that generalizes the stretched exponential function. Its generating differential equation is then presented. In Section 5, we consider the pdfs for continuous variables. First we consider the one-tail stretched exponential generalization and obtain analytically its cumulative function and moments. One obtains the generalized error function as a particular case and the Zipf-Mandelbrot pdf as another. Next, we consider the two-tail generalized stretched exponential pdf and obtain analytically its cumulative function and moments. One has the generalized Gaussian and the generalized Laplace pdf as particular cases. The characteristic function is analytically calculated. Our final remarks are drawn in Section 6.


## 2. Theq̃-Generalized Functions

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 [9, 10]. Inverting this function, one obtains the generalized exponential function. Some properties of these generalized functions are presented.


### 2.1.q̃-Generalized Logarithm Function

In the one-parameter generalization we address here, theq̃-logarithm function lnq̃(x) is defined as the value of the area underneath 1/t1-q̃, in the interval t∈[1,x] [29]
(2.1)lnq̃(x)=∫1xdtt1-q̃=limq̃′→q̃xq̃′-1q̃′,
which is defined for all real values of q̃, but only for positive x. This is exactly the same function obtained from the nonextensive statistical mechanics context [8, 9], but here, only simple geometrical arguments have been used in the derivation.The usual natural logarithm (lnx) is retrieved for q̃=0 and a linear function for q̃=1. Scaling and deformation of the variable x are given by lnq̃(αxβ)=βlnβq̃(α1/βx), so that for β=-1, one has lnq̃(α/x)=-ln-q̃(x/α) and for the particular case α=1, lnq̃(x-1)=-ln-q̃(x). Notice that using q̃ instead of q=1-q̃, as used in [8, 9], makes simpler to handle the scaling and deformation operations.


### 2.2.q̃-Generalized Exponential Function

Theq̃-exponential function eq̃(x) is defined as the t-value, in such a way that the area underneath fq̃(t)=1/t1-q̃, in the interval t∈[1,eq̃(x)], is x. This is the inverse of the q̃-logarithm function eq̃[lnq̃(x)]=x=lnq̃[eq̃(x)] and it is given by
(2.2)eq̃(x)={0,forq̃x<-1,limq̃′→q̃(1+q̃′x)1/q̃′,forq̃x≥-1.
Notice that (1+q̃x)1/q̃ is real only if q̃x≥-1.This is a nonnegative function (eq̃(x)≥0, for all q̃ and x) and eq̃(0)=1, independently of the q̃ value. For q̃=0, one retrieves the usual exponential function ex and for q̃=1, a linear function. Notice that taking the surface underneath 1/t1-q̃ to be unitary, one generalizes the Euler’s number: eq̃=eq̃(1)=(1+q̃)1/q̃. An interesting property is that [eq̃(x)]a=eq̃/a(ax), meaning that the q̃-exponential argument scaling corresponds to a power of a different q̃-exponential function. For a=-1, one has that e-q̃(-x)=1/eq̃(x).The derivative of theq̃-exponential function with respect to x is deq̃(kx)/dx=k[eq̃(kx)]1-q̃=keq̃/(1-q̃)[(1-q̃)kx], so that it is the solution of the following nonlinear first-order differential equation: dy(x)/dx=ky1-q̃(x), which is a particular case of Bernoulli’s differential equation: [d/dx+p(x)]y(x)=q(x)y1-q̃(x), with p(x)=0 and q(x)=k. Notice that k has the dimension of yq̃ over the dimension of x. This means that it sets up a scale (inversion dimension of x) to the problem only if q̃=0. One important application of the above differential equation concerns reaction kinetics [30].


## 2.1.q̃-Generalized Logarithm Function

In the one-parameter generalization we address here, theq̃-logarithm function lnq̃(x) is defined as the value of the area underneath 1/t1-q̃, in the interval t∈[1,x] [29]
(2.1)lnq̃(x)=∫1xdtt1-q̃=limq̃′→q̃xq̃′-1q̃′,
which is defined for all real values of q̃, but only for positive x. This is exactly the same function obtained from the nonextensive statistical mechanics context [8, 9], but here, only simple geometrical arguments have been used in the derivation.The usual natural logarithm (lnx) is retrieved for q̃=0 and a linear function for q̃=1. Scaling and deformation of the variable x are given by lnq̃(αxβ)=βlnβq̃(α1/βx), so that for β=-1, one has lnq̃(α/x)=-ln-q̃(x/α) and for the particular case α=1, lnq̃(x-1)=-ln-q̃(x). Notice that using q̃ instead of q=1-q̃, as used in [8, 9], makes simpler to handle the scaling and deformation operations.


## 2.2.q̃-Generalized Exponential Function

Theq̃-exponential function eq̃(x) is defined as the t-value, in such a way that the area underneath fq̃(t)=1/t1-q̃, in the interval t∈[1,eq̃(x)], is x. This is the inverse of the q̃-logarithm function eq̃[lnq̃(x)]=x=lnq̃[eq̃(x)] and it is given by
(2.2)eq̃(x)={0,forq̃x<-1,limq̃′→q̃(1+q̃′x)1/q̃′,forq̃x≥-1.
Notice that (1+q̃x)1/q̃ is real only if q̃x≥-1.This is a nonnegative function (eq̃(x)≥0, for all q̃ and x) and eq̃(0)=1, independently of the q̃ value. For q̃=0, one retrieves the usual exponential function ex and for q̃=1, a linear function. Notice that taking the surface underneath 1/t1-q̃ to be unitary, one generalizes the Euler’s number: eq̃=eq̃(1)=(1+q̃)1/q̃. An interesting property is that [eq̃(x)]a=eq̃/a(ax), meaning that the q̃-exponential argument scaling corresponds to a power of a different q̃-exponential function. For a=-1, one has that e-q̃(-x)=1/eq̃(x).The derivative of theq̃-exponential function with respect to x is deq̃(kx)/dx=k[eq̃(kx)]1-q̃=keq̃/(1-q̃)[(1-q̃)kx], so that it is the solution of the following nonlinear first-order differential equation: dy(x)/dx=ky1-q̃(x), which is a particular case of Bernoulli’s differential equation: [d/dx+p(x)]y(x)=q(x)y1-q̃(x), with p(x)=0 and q(x)=k. Notice that k has the dimension of yq̃ over the dimension of x. This means that it sets up a scale (inversion dimension of x) to the problem only if q̃=0. One important application of the above differential equation concerns reaction kinetics [30].


## 3. Beta-Like Distribution

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 [28] is conveniently described by the q̃-exponential function. In this way, we are able to quantify the finite size effects. This rank distribution is very reliable since it has an underlying microscopic model and has been validated by a wide range of experimental data [28].To simultaneously fit the beginning, body and tail of experimental rank distributions of complex systems, Naumis and Cocho [28] consider N independent subsystems with a large number of internal states. The rank r of a system property dependent on the internal states of the subsystems decays as a two-free-parameter beta-like function: f(r)=K[1-r/(R+1)]βr-α, where K=1/∑r=1N[1-r/(R+1)]βr-α, with R≤N being the maximal value of r, K is the normalization factor, and the two free parameters are α and β. As noticed by the authors, if R≫1, then K≈1/B(1-α,1+β), where B(a,b)=Γ(a)Γ(b)/Γ(a+b) is the beta function [31] and Γ(x)=∫0∞dttx-1e-t, x>0 is the gamma function [31].Finite size effects are described by factor[1-r/(R+1)]β≈1-βr/R≈e-r/r0, for R≫r with r0=R/β. In this way, we see that the rank distribution of f(r) is in fact a generalization of the standard technique of multiplying the power-law (r-α) by an exponential cutoff: f(r)∝r-αe-r/r0.If one writes[1-r/(R+1)]β=[1-(1/β)βr/(R+1)]β=e1/β[-βr/(R+1)] then
(3.1)f(r)=Krαe1/β[-βrR+1]=Krα[e1(-rR+1)]β,
meaning that the q̃-exponential function can properly take into account finite size effects. The exponential cutoff is retrieved where R and β grow, but R must grow faster than β, to have R→∞ and β→∞, with β/R→0.


## 4. Special Functions and Processes

In what follows, we first show that the Zipf-Mandelbrot function, which is a fingerprint of complex systems, can be written in terms of theq̃-exponential. Next, raising the q̃-exponential argument to a given power, one obtains a function that generalizes the stretched exponential function. Finally, we obtain the process (differential equation) of which the generalized stretched exponential is the solution.


### 4.1. Zipf-Mandelbrot Function

The envelope of a typical rank distribution of complex systems can be well described by the Zipf-Mandelbrot function [32], which may be written in terms of q̃-exponential function:
(4.1)Pq̃,α,A(x)=d(c+x)γ=Aeq̃(x/α)=Ae-q̃(-xα),
with q̃=1/γ, α=c/γ, and A=d/cγ.We remark that (4.1) has another interesting application in time-dependent luminescence spectroscopy. In this case the relaxation processes are known as Becquerel decay function [33].


### 4.2. Generalized Stretched Exponential Function

A deformation in the argument of (4.1) leads to the generalized stretched exponential function
(4.2)Pq̃,α,β,A(x)=Pq̃,α,A(x1/β)=d(d+x1/β)γ=Ae-q̃(-x1/βα).The usual stretched exponential function, also known as the Kohlrausch function [34–36], is obtained from (4.2) in the limit q̃→0. Although the (usual) stretched exponential function has been used to describe relaxation processes in time-dependent luminescence spectroscopy [36], a generalization of the form eβ/ae-e1/β(x/α)/a seems to be more convenient to fit experimetal data [37]. In this case we stress the q̃-exponential figures as the argument of the usual exponential function.The stretched exponential function can also be obtained as the solution of the following differential equation:dy(x)/dx=-x1/β-1y(x)/(αβ), which can be written in terms of the logarithmic function (relative variation) as dlny(x)/dx=-x1/β-1/(αβ). If one replaces the logarithmic function in the differential equation obtained above by the q̃-logarithm, one obtains
(4.3)dln-q̃y(x)dx=-x1/β-1αβ,
or equivalently
(4.4)dy(x)dx=-x1/β-1αβy1+q̃(x),
which is a particular case of Bernoulli’s equation: dy(x)/dx+p(x)y(x)=q(x)y1+q̃(x), with p(x)=0 and q(x)=-x1/β-1/(αβ) and which solution is precisely the generalized stretched exponential function of (4.2).


## 4.1. Zipf-Mandelbrot Function

The envelope of a typical rank distribution of complex systems can be well described by the Zipf-Mandelbrot function [32], which may be written in terms of q̃-exponential function:
(4.1)Pq̃,α,A(x)=d(c+x)γ=Aeq̃(x/α)=Ae-q̃(-xα),
with q̃=1/γ, α=c/γ, and A=d/cγ.We remark that (4.1) has another interesting application in time-dependent luminescence spectroscopy. In this case the relaxation processes are known as Becquerel decay function [33].


## 4.2. Generalized Stretched Exponential Function

A deformation in the argument of (4.1) leads to the generalized stretched exponential function
(4.2)Pq̃,α,β,A(x)=Pq̃,α,A(x1/β)=d(d+x1/β)γ=Ae-q̃(-x1/βα).The usual stretched exponential function, also known as the Kohlrausch function [34–36], is obtained from (4.2) in the limit q̃→0. Although the (usual) stretched exponential function has been used to describe relaxation processes in time-dependent luminescence spectroscopy [36], a generalization of the form eβ/ae-e1/β(x/α)/a seems to be more convenient to fit experimetal data [37]. In this case we stress the q̃-exponential figures as the argument of the usual exponential function.The stretched exponential function can also be obtained as the solution of the following differential equation:dy(x)/dx=-x1/β-1y(x)/(αβ), which can be written in terms of the logarithmic function (relative variation) as dlny(x)/dx=-x1/β-1/(αβ). If one replaces the logarithmic function in the differential equation obtained above by the q̃-logarithm, one obtains
(4.3)dln-q̃y(x)dx=-x1/β-1αβ,
or equivalently
(4.4)dy(x)dx=-x1/β-1αβy1+q̃(x),
which is a particular case of Bernoulli’s equation: dy(x)/dx+p(x)y(x)=q(x)y1+q̃(x), with p(x)=0 and q(x)=-x1/β-1/(αβ) and which solution is precisely the generalized stretched exponential function of (4.2).


## 5. Probability Functions

Considering the factorA>0 and using the nonnegativeness of (4.2), we write the generalization of the stretched exponential pdf and study its properties. We consider one- and two-tail distributions and obtain some known pdfs (generalized Gaussian) and new ones (generalized error function and generalized Laplace pdf) as particular cases.


### 5.1. One-Tail PDF

If the considered independent variablex is constrained to nonnegative (or eventually nonpositive) values, then one uses the one-tail pdf. We consider the generalization of the stretched exponential pdf and analytically obtain its cumulative function and moments. From this pdf, one obtains the generalized error function as a particular case and the Zipf-Mandelbrot pdf as another.The normalization factorA of (4.2) is
(5.1)1A=∫0∞dte-q̃(-t1/βα)=β(αq̃)βB(β,1q̃-β).The integral of (5.1) does not diverge only if 0<q̃<1, 0<β<1/q̃, α>0, and one has the generalized stretched exponential pdf:
(5.2)Pq̃,α,β(1)(x)=(q̃/α)ββB(β,1/q̃-β)e-q̃[-x1/βα].For our purposes, it is more convenient to writea=(αβ)β>0:
(5.3)Pq̃,a,β(1)(x)=(q̃β)βaβB(β,1/q̃-β)e-q̃[-β(xa)1/β],
which is depicted in Figures 1 and 2.Behavior of (5.3) as a function of  (a) x and q̃, with β=1/2 and a=1, (b) x and β, with q̃=1/5 and a=1, (c) q̃ and β, with x=1 and a=1.
(a)(b)(c)Behavior of (5.3) as a function of (a) x with β=1/2 and a=1 and for several q̃ values (projection of Figure 1(a)), (b) x with q̃=1/5 and a=1 and for several β values (projection of Figure 1(b)).
(a)(b)Asq̃→0, B(β,1/q̃-β)≈q̃βΓ(β), since 1/q̃≫β, one retrieves the stretched exponential function P0,a,β(1)(x)=ββe-β(x/a)1/β/[aβΓ(β)] or P0,α,β(1)(x)=e-x1/β/α/[αββΓ(β)].The cumulative function of (5.3) is
(5.4)Fq̃,a,β(1)(x)=∫0xdtPq̃,a,β(1)(t)=xA2F1(β,1q;β+1;-q̃x1/βα),
where
(5.5)2F1(a,b;c;x)=∑n=0∞(a)n(b)n(c)nxnn!is the hypergeometric function [31] and
(5.6)(a)n=a(a+1)(a+2)⋯(a+n-1)=(a+n-1)!(a-1)!=Γ(a+n)Γ(a)is the Pochhammer symbol.The moments ofx are
(5.7)〈xn〉q̃,a,β=[a(q̃β)β]nB[(n+1)β,1/q̃-(n+1)β]B(β,1/q̃-β),
where one sees that they are finite only if q̃<1/[(n+1)β].Ifq̃<1/(3β), the mean value and variance are finite and, respectively, given by
(5.8)〈x〉q̃,a,β=(αq̃)βB(2β,1/q̃-2β)B(β,1/q̃-β),σq̃,a,β2〈x〉q̃,a,β2=B(3β,1/q̃-3β)B(β,1/q̃-β)B2(2β,1/q̃-2β)-1.
Notice that the ratio σq̃,a,β2/〈x〉q̃,a,β2 depends only on β and q̃, but not on a.Particular values ofβ lead to a q̃-generalization of the error function and to the Zipf-Mandelbrot pdf.


#### 5.1.1. Generalized Error Function

To generalize the error function, considerβ=1/2 and a=ββ (or α=1) in (5.4) and one has
(5.9)erfq̃(x)=Fq̃,1/2,1/2(1)(x)=2q̃B(1/2,1/q̃-1/2)∫0xdte-q̃(-t2).
As q̃→0, B(1/2,1/q̃-1/2)=q̃π and one retrieves the standard error function: erf0(x)=erf(x)=(2/π)∫0xdte-t2.An alternative way to obtain (5.9) can be found in [38].


#### 5.1.2. Zipf-Mandelbrot PDF

Forβ=1 in (5.2), one obtains the Zipf-Mandelbrot’s pdf
(5.10)Pq̃,α,1(1)(x)=1-q̃αe-q̃(-xα)=1-q̃α1(1+q̃x/α)1/q̃,
where the mean value and the variance are
(5.11)〈x〉q̃,α,1=α1-2q̃,σq̃,α,12〈x〉q̃,α,12=q̃-13q̃-1,
which is finite for 0≤q̃<1/3 and, from (5.4), one obtains its cumulative function:
(5.12)Fq̃,a,1(1)(x)=1-1[eq̃(x/a)]1-q̃=1-F̃q̃,a,1(1)(x),
where the upper-tail distribution is simply given by
(5.13)F̃q̃,a,1(x)=∫x∞dtPq̃,a,1(1)(t)=(1+q̃xa)e-q̃(-xa)=[e-q̃(-xa)]1-q̃=[eq̃(xa)]q̃-1,
which is more suitable for fitting the model to real data than the pdf (5.10) itself.


### 5.2. Two-Tail PDF

If the domain of the considered independent variable is not bounded, it is interesting to consider its absolute value|x| in (5.3) and one has a symmetric pdf about the line x=0. Notice that in this case, the normalization factor must be halved since the domain has been doubled in a symmetrical way, the generalized stretched exponential pdf is then
(5.14)Pq̃,a,β(2)(x)=(q̃β)β2aβB(β,1/q̃-β)e-q̃[-β(|x|a)1/β],
which is a convenient function to use in wavelets [39]. Its cumulative function is
(5.15)Fq̃,a,β(2)(x)=12[1+sgn(x)Fq̃,a,β(1)(|x|)],
where Fq̃,a,β(1)(x) is given by (5.4).On one hand, due to its symmetry aroundx=0, the odd moments of this pdf vanish 〈x2n+1〉=0, with n=0,1,2,…. On the other hand, the even moments 〈x2n〉 are finite only if q̃<1/[(2n+1)β]:
(5.16)〈x2n〉q̃,a,β=[a(q̃β)β]2nB[(1+2n)β,1/q̃-(1+2n)β]B(β,1/q̃-β).In the following, we retrieve the generalized Gaussian as a particular case of (5.14). Also, as a new result, we propose to consider another particular case of (5.14) to generalize the Laplacian pdf. The characteristic function of both particular cases is analytically calculated.


#### 5.2.1. Generalized Gaussian

An interesting particular case of the two-tail generalized stretched exponential function (5.14) is when β=1/2, which leads to the q̃-Gaussian [40]
(5.17)Gq̃,a(x)=q̃Γ(1/q̃)Γ(1/q̃-1/2)e-q̃[-(x/a)2/2]2πa2.
Due to symmetry, all odd moments vanish. Even moments are given by (5.16)
(5.18)〈x2n〉q̃,a,1/2=(2a2q̃)nΓ(1/2+n)Γ(1/q̃-1/2-n)πΓ(1/q̃-1/2),
and the variance is finite only for q̃<2/3(5.19)σq̃,a2=〈x2〉q̃,a=2a22-3q̃.Using that fora≫b, Γ(a+b)/Γ(a)=ab, when q̃→0 in (5.17), Γ(1/q̃)/Γ(1/q̃-1/2)=1/q̃1/2 and one has a Guassian, with variance a2: G0,a(x)=e-(x/a)2/2/2πa2, as a particular case. Another particular case is when q̃=1 and one retrieves the Lorentzian (Cauchy pdf) G1,a(x)=1/{πa2[1+(x/a)2/2]}.The characteristic function of (5.17) has an analytical closed form
(5.20)pq̃,a,1/2(k)=〈eıkx〉=∫-∞∞dxGq̃,a(x)eıkx=(ka2/q̃2)1/q̃-1/22K1/q̃-1/2(ka2/q̃)Γ(1/q̃-1/2),
where Kν(z) is the K-modified Bessel function [31]:
(5.21)Kν(z)=2νΓ(ν+1/2)πzν+1∫0∞dtcost[1+(t/z)2]ν+1/2.
For the Gaussian one has also a Gaussian p0,a(k)=e-a2k2/2 but for the Lorentzian, one has the Laplace function p1,a=e-2a|k|.


#### 5.2.2. Generalized Laplace PDF

Forβ=1 and q̃=0, (5.14) leads to the Laplace pdf P0,a,1(2)(x)=e-|x|/a/(2a). For arbitrary q̃, one has the generalized Laplace pdf(5.22)Lq̃,a(x)=Pq̃,a,1(2)(x)=1-q̃2ae-q̃(-|x|a),
and its cumulative function is
(5.23)Fq̃,a,1(2)(x)=12P̃q̃,a,1(|x|),
where P̃q̃,a,1(x) is given by (5.10).The odd moments of (5.22) vanish and the even ones are finite if q̃<1/(1+2n)(5.24)〈x2n〉q̃,a,1=1q̃-1(aq̃)2nB(1+2n,1q̃-2n-1).
Its characteristic function has an analytical closed form
(5.25)pq̃,a,1(k)=∫-∞∞dxLq̃,a(x)eıkx=1-q̃a∫0∞dxcos(kx)e-q̃(-xa)=1-q̃a∫0∞dxcos(kx)(1+q̃x/a)1/q̃=πt01/q̃-1sin(t0+π/(2q̃))sin(π/q̃)Γ(1/q̃-1)+1F2(1;1-12q̃,32-12q̃;-t024),
where t0=ka/q̃ and 1F2(a;b1,b2;z)=∑n=0∞{(a)n/[(b1)n(b2)n]}xn/n! is the hypergeometric function [31], with (a)n being the Pochhammer symbol.


## 5.1. One-Tail PDF

If the considered independent variablex is constrained to nonnegative (or eventually nonpositive) values, then one uses the one-tail pdf. We consider the generalization of the stretched exponential pdf and analytically obtain its cumulative function and moments. From this pdf, one obtains the generalized error function as a particular case and the Zipf-Mandelbrot pdf as another.The normalization factorA of (4.2) is
(5.1)1A=∫0∞dte-q̃(-t1/βα)=β(αq̃)βB(β,1q̃-β).The integral of (5.1) does not diverge only if 0<q̃<1, 0<β<1/q̃, α>0, and one has the generalized stretched exponential pdf:
(5.2)Pq̃,α,β(1)(x)=(q̃/α)ββB(β,1/q̃-β)e-q̃[-x1/βα].For our purposes, it is more convenient to writea=(αβ)β>0:
(5.3)Pq̃,a,β(1)(x)=(q̃β)βaβB(β,1/q̃-β)e-q̃[-β(xa)1/β],
which is depicted in Figures 1 and 2.Behavior of (5.3) as a function of  (a) x and q̃, with β=1/2 and a=1, (b) x and β, with q̃=1/5 and a=1, (c) q̃ and β, with x=1 and a=1.
(a)(b)(c)Behavior of (5.3) as a function of (a) x with β=1/2 and a=1 and for several q̃ values (projection of Figure 1(a)), (b) x with q̃=1/5 and a=1 and for several β values (projection of Figure 1(b)).
(a)(b)Asq̃→0, B(β,1/q̃-β)≈q̃βΓ(β), since 1/q̃≫β, one retrieves the stretched exponential function P0,a,β(1)(x)=ββe-β(x/a)1/β/[aβΓ(β)] or P0,α,β(1)(x)=e-x1/β/α/[αββΓ(β)].The cumulative function of (5.3) is
(5.4)Fq̃,a,β(1)(x)=∫0xdtPq̃,a,β(1)(t)=xA2F1(β,1q;β+1;-q̃x1/βα),
where
(5.5)2F1(a,b;c;x)=∑n=0∞(a)n(b)n(c)nxnn!is the hypergeometric function [31] and
(5.6)(a)n=a(a+1)(a+2)⋯(a+n-1)=(a+n-1)!(a-1)!=Γ(a+n)Γ(a)is the Pochhammer symbol.The moments ofx are
(5.7)〈xn〉q̃,a,β=[a(q̃β)β]nB[(n+1)β,1/q̃-(n+1)β]B(β,1/q̃-β),
where one sees that they are finite only if q̃<1/[(n+1)β].Ifq̃<1/(3β), the mean value and variance are finite and, respectively, given by
(5.8)〈x〉q̃,a,β=(αq̃)βB(2β,1/q̃-2β)B(β,1/q̃-β),σq̃,a,β2〈x〉q̃,a,β2=B(3β,1/q̃-3β)B(β,1/q̃-β)B2(2β,1/q̃-2β)-1.
Notice that the ratio σq̃,a,β2/〈x〉q̃,a,β2 depends only on β and q̃, but not on a.Particular values ofβ lead to a q̃-generalization of the error function and to the Zipf-Mandelbrot pdf.


### 5.1.1. Generalized Error Function

To generalize the error function, considerβ=1/2 and a=ββ (or α=1) in (5.4) and one has
(5.9)erfq̃(x)=Fq̃,1/2,1/2(1)(x)=2q̃B(1/2,1/q̃-1/2)∫0xdte-q̃(-t2).
As q̃→0, B(1/2,1/q̃-1/2)=q̃π and one retrieves the standard error function: erf0(x)=erf(x)=(2/π)∫0xdte-t2.An alternative way to obtain (5.9) can be found in [38].


### 5.1.2. Zipf-Mandelbrot PDF

Forβ=1 in (5.2), one obtains the Zipf-Mandelbrot’s pdf
(5.10)Pq̃,α,1(1)(x)=1-q̃αe-q̃(-xα)=1-q̃α1(1+q̃x/α)1/q̃,
where the mean value and the variance are
(5.11)〈x〉q̃,α,1=α1-2q̃,σq̃,α,12〈x〉q̃,α,12=q̃-13q̃-1,
which is finite for 0≤q̃<1/3 and, from (5.4), one obtains its cumulative function:
(5.12)Fq̃,a,1(1)(x)=1-1[eq̃(x/a)]1-q̃=1-F̃q̃,a,1(1)(x),
where the upper-tail distribution is simply given by
(5.13)F̃q̃,a,1(x)=∫x∞dtPq̃,a,1(1)(t)=(1+q̃xa)e-q̃(-xa)=[e-q̃(-xa)]1-q̃=[eq̃(xa)]q̃-1,
which is more suitable for fitting the model to real data than the pdf (5.10) itself.


## 5.1.1. Generalized Error Function

To generalize the error function, considerβ=1/2 and a=ββ (or α=1) in (5.4) and one has
(5.9)erfq̃(x)=Fq̃,1/2,1/2(1)(x)=2q̃B(1/2,1/q̃-1/2)∫0xdte-q̃(-t2).
As q̃→0, B(1/2,1/q̃-1/2)=q̃π and one retrieves the standard error function: erf0(x)=erf(x)=(2/π)∫0xdte-t2.An alternative way to obtain (5.9) can be found in [38].


## 5.1.2. Zipf-Mandelbrot PDF

Forβ=1 in (5.2), one obtains the Zipf-Mandelbrot’s pdf
(5.10)Pq̃,α,1(1)(x)=1-q̃αe-q̃(-xα)=1-q̃α1(1+q̃x/α)1/q̃,
where the mean value and the variance are
(5.11)〈x〉q̃,α,1=α1-2q̃,σq̃,α,12〈x〉q̃,α,12=q̃-13q̃-1,
which is finite for 0≤q̃<1/3 and, from (5.4), one obtains its cumulative function:
(5.12)Fq̃,a,1(1)(x)=1-1[eq̃(x/a)]1-q̃=1-F̃q̃,a,1(1)(x),
where the upper-tail distribution is simply given by
(5.13)F̃q̃,a,1(x)=∫x∞dtPq̃,a,1(1)(t)=(1+q̃xa)e-q̃(-xa)=[e-q̃(-xa)]1-q̃=[eq̃(xa)]q̃-1,
which is more suitable for fitting the model to real data than the pdf (5.10) itself.


## 5.2. Two-Tail PDF

If the domain of the considered independent variable is not bounded, it is interesting to consider its absolute value|x| in (5.3) and one has a symmetric pdf about the line x=0. Notice that in this case, the normalization factor must be halved since the domain has been doubled in a symmetrical way, the generalized stretched exponential pdf is then
(5.14)Pq̃,a,β(2)(x)=(q̃β)β2aβB(β,1/q̃-β)e-q̃[-β(|x|a)1/β],
which is a convenient function to use in wavelets [39]. Its cumulative function is
(5.15)Fq̃,a,β(2)(x)=12[1+sgn(x)Fq̃,a,β(1)(|x|)],
where Fq̃,a,β(1)(x) is given by (5.4).On one hand, due to its symmetry aroundx=0, the odd moments of this pdf vanish 〈x2n+1〉=0, with n=0,1,2,…. On the other hand, the even moments 〈x2n〉 are finite only if q̃<1/[(2n+1)β]:
(5.16)〈x2n〉q̃,a,β=[a(q̃β)β]2nB[(1+2n)β,1/q̃-(1+2n)β]B(β,1/q̃-β).In the following, we retrieve the generalized Gaussian as a particular case of (5.14). Also, as a new result, we propose to consider another particular case of (5.14) to generalize the Laplacian pdf. The characteristic function of both particular cases is analytically calculated.


### 5.2.1. Generalized Gaussian

An interesting particular case of the two-tail generalized stretched exponential function (5.14) is when β=1/2, which leads to the q̃-Gaussian [40]
(5.17)Gq̃,a(x)=q̃Γ(1/q̃)Γ(1/q̃-1/2)e-q̃[-(x/a)2/2]2πa2.
Due to symmetry, all odd moments vanish. Even moments are given by (5.16)
(5.18)〈x2n〉q̃,a,1/2=(2a2q̃)nΓ(1/2+n)Γ(1/q̃-1/2-n)πΓ(1/q̃-1/2),
and the variance is finite only for q̃<2/3(5.19)σq̃,a2=〈x2〉q̃,a=2a22-3q̃.Using that fora≫b, Γ(a+b)/Γ(a)=ab, when q̃→0 in (5.17), Γ(1/q̃)/Γ(1/q̃-1/2)=1/q̃1/2 and one has a Guassian, with variance a2: G0,a(x)=e-(x/a)2/2/2πa2, as a particular case. Another particular case is when q̃=1 and one retrieves the Lorentzian (Cauchy pdf) G1,a(x)=1/{πa2[1+(x/a)2/2]}.The characteristic function of (5.17) has an analytical closed form
(5.20)pq̃,a,1/2(k)=〈eıkx〉=∫-∞∞dxGq̃,a(x)eıkx=(ka2/q̃2)1/q̃-1/22K1/q̃-1/2(ka2/q̃)Γ(1/q̃-1/2),
where Kν(z) is the K-modified Bessel function [31]:
(5.21)Kν(z)=2νΓ(ν+1/2)πzν+1∫0∞dtcost[1+(t/z)2]ν+1/2.
For the Gaussian one has also a Gaussian p0,a(k)=e-a2k2/2 but for the Lorentzian, one has the Laplace function p1,a=e-2a|k|.


### 5.2.2. Generalized Laplace PDF

Forβ=1 and q̃=0, (5.14) leads to the Laplace pdf P0,a,1(2)(x)=e-|x|/a/(2a). For arbitrary q̃, one has the generalized Laplace pdf(5.22)Lq̃,a(x)=Pq̃,a,1(2)(x)=1-q̃2ae-q̃(-|x|a),
and its cumulative function is
(5.23)Fq̃,a,1(2)(x)=12P̃q̃,a,1(|x|),
where P̃q̃,a,1(x) is given by (5.10).The odd moments of (5.22) vanish and the even ones are finite if q̃<1/(1+2n)(5.24)〈x2n〉q̃,a,1=1q̃-1(aq̃)2nB(1+2n,1q̃-2n-1).
Its characteristic function has an analytical closed form
(5.25)pq̃,a,1(k)=∫-∞∞dxLq̃,a(x)eıkx=1-q̃a∫0∞dxcos(kx)e-q̃(-xa)=1-q̃a∫0∞dxcos(kx)(1+q̃x/a)1/q̃=πt01/q̃-1sin(t0+π/(2q̃))sin(π/q̃)Γ(1/q̃-1)+1F2(1;1-12q̃,32-12q̃;-t024),
where t0=ka/q̃ and 1F2(a;b1,b2;z)=∑n=0∞{(a)n/[(b1)n(b2)n]}xn/n! is the hypergeometric function [31], with (a)n being the Pochhammer symbol.


## 5.2.1. Generalized Gaussian

An interesting particular case of the two-tail generalized stretched exponential function (5.14) is when β=1/2, which leads to the q̃-Gaussian [40]
(5.17)Gq̃,a(x)=q̃Γ(1/q̃)Γ(1/q̃-1/2)e-q̃[-(x/a)2/2]2πa2.
Due to symmetry, all odd moments vanish. Even moments are given by (5.16)
(5.18)〈x2n〉q̃,a,1/2=(2a2q̃)nΓ(1/2+n)Γ(1/q̃-1/2-n)πΓ(1/q̃-1/2),
and the variance is finite only for q̃<2/3(5.19)σq̃,a2=〈x2〉q̃,a=2a22-3q̃.Using that fora≫b, Γ(a+b)/Γ(a)=ab, when q̃→0 in (5.17), Γ(1/q̃)/Γ(1/q̃-1/2)=1/q̃1/2 and one has a Guassian, with variance a2: G0,a(x)=e-(x/a)2/2/2πa2, as a particular case. Another particular case is when q̃=1 and one retrieves the Lorentzian (Cauchy pdf) G1,a(x)=1/{πa2[1+(x/a)2/2]}.The characteristic function of (5.17) has an analytical closed form
(5.20)pq̃,a,1/2(k)=〈eıkx〉=∫-∞∞dxGq̃,a(x)eıkx=(ka2/q̃2)1/q̃-1/22K1/q̃-1/2(ka2/q̃)Γ(1/q̃-1/2),
where Kν(z) is the K-modified Bessel function [31]:
(5.21)Kν(z)=2νΓ(ν+1/2)πzν+1∫0∞dtcost[1+(t/z)2]ν+1/2.
For the Gaussian one has also a Gaussian p0,a(k)=e-a2k2/2 but for the Lorentzian, one has the Laplace function p1,a=e-2a|k|.


## 5.2.2. Generalized Laplace PDF

Forβ=1 and q̃=0, (5.14) leads to the Laplace pdf P0,a,1(2)(x)=e-|x|/a/(2a). For arbitrary q̃, one has the generalized Laplace pdf(5.22)Lq̃,a(x)=Pq̃,a,1(2)(x)=1-q̃2ae-q̃(-|x|a),
and its cumulative function is
(5.23)Fq̃,a,1(2)(x)=12P̃q̃,a,1(|x|),
where P̃q̃,a,1(x) is given by (5.10).The odd moments of (5.22) vanish and the even ones are finite if q̃<1/(1+2n)(5.24)〈x2n〉q̃,a,1=1q̃-1(aq̃)2nB(1+2n,1q̃-2n-1).
Its characteristic function has an analytical closed form
(5.25)pq̃,a,1(k)=∫-∞∞dxLq̃,a(x)eıkx=1-q̃a∫0∞dxcos(kx)e-q̃(-xa)=1-q̃a∫0∞dxcos(kx)(1+q̃x/a)1/q̃=πt01/q̃-1sin(t0+π/(2q̃))sin(π/q̃)Γ(1/q̃-1)+1F2(1;1-12q̃,32-12q̃;-t024),
where t0=ka/q̃ and 1F2(a;b1,b2;z)=∑n=0∞{(a)n/[(b1)n(b2)n]}xn/n! is the hypergeometric function [31], with (a)n being the Pochhammer symbol.


## 6. Conclusion

We have shown that theq̃-generalization of the exponential is suitable to generalize the stretched exponential function. The q̃-generalized stretched exponential function has the generalized error function, the generalized Laplace pdf and the already known generalized Gaussian as special cases. Further, we have used the q̃-exponential to write the very reliable rank distribution obtained by Naumis and Cocho. Since these distributions are the solution of differential equations that describe the complex systems, the q̃-generalization brings many different systems to be described by the same underlying process.



---

*Source: 206176-2010-01-05.xml*