The Wright Functions of the Second Kind in Mathematical Physics

Abstract

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.

1. Introduction

The special functions play a fundamental role in all fields of Applied Mathematics and Mathematical Physics because any analytical results are expressed in terms of some of these functions. Even if the topic of special functions can appear boring and their properties mainly treated in handbooks, we would promote the relevance of some of them not yet so well known. We devote our attention to the Wright functions, in particular with the class of the second kind. These functions, as we will see hereafter, are fundamental to deal with some non-standard deterministic and stochastic processes. Indeed, the Gaussian function (known as the normal probability distribution) must be generalized in a suitable way in the framework of partial differential equations of non-integer order for describing the anomalous diffusion and the transition from fractional diffusion to wave propagation.

Furthermore, their usefulness and meaningfulness also extends to other topics. For example, these functions and their Laplace Transforms can be applied in electromagnetic problems, see the 1958 paper by Ragab [1] (where the Wright functions were used without knowing their existence) and the recent 2020 paper by Stefański and Gulgowski [2]. Recently, the Wright functions have been used in the theory of coherent states by Garra, Giraldi, and Mainardi [3].

This survey article aims to discuss the relevance of the Wright Functions and also to focus on the not well-known Four Sisters Functions and their importance in time-fractional diffusion-wave equations.

The plan of the paper is organized as follows. In Section 2, we introduce the Wright functions, entirely in the complex plane that we distinguish in two kinds in relation to the value-range of the two parameters on which they depend. In particular, we devote our attention to two Wright functions of the second kind introduced by Mainardi with the term of auxiliary functions. One of them, known as M-Wright function, generalizes the Gaussian function so it is expected to play a fundamental role in non-Gaussian stochastic processes.

Indeed, in Section 3, we show how the Wright functions of the second kind are relevant in the analysis of time-fractional diffusion and diffusion-wave equations being related to their fundamental solutions. This analysis leads to generalizing the known results r of the standard diffusion equation in the one-dimensional case that is recalled in Appendix A by means of auxiliary functions as particular cases of the Wright functions of the second kind known as M-Wright or Mainardi functions. For readers’ convenience, in Appendix B, we will also provide an introduction to the time-derivative of fractional order in the Caputo sense We remind that nowadays, as usual, by fractional order, we mean a non-integer order, so that the term “fractional” is a misnomer kept only for historical reasons.

In Section 4, we consider again the Mainardi auxiliary functions functions for their role in probability theory and in particular in the framework of Lévy stable distributions whose general theory is recalled in Appendix C.

In Section 5, we show how the auxiliary functions turn out to be included in a class that we denote the four sister functions. On their turn, these four functions depending on a real parameter $\nu \in (0,1)$ are the natural generalization of the three sisters functions introduced in Appendix A devoted to the standard diffusion equation. The attribute of sisters was put in by one of us (F. M.) because of their inter-relations, in his lecture notes on Mathematical Physics, so this is only a personal reason that we hope to be shared by the readers.

Finally, in Section 6, we provide some concluding remarks paying attention to work to be done in the next future.

We point out that we have equipped our theoretical analysis with several plots hoping they will be considered illuminating for the interested readers. We also note that we have limited our review to the simplest boundary values problems of equations in one space dimension referring the readers to suitable references for more general treatments in Section 3.1.

2. The Wright Functions of the Second Kind and the Mainardi Auxiliary Functions

The classical Wright function that we denote by $W_{\lambda ,\mu }\left(z\right)$, is defined by the series representation convergent in the whole complex plane,

$$W_{\lambda ,\mu }\left(z\right):=\sum _{n=0}^{\infty }\frac{z^n}{n!\mathsf{\Gamma }(\lambda n+\mu )},\lambda >-1,\mu \in \mathbb{C},$$

(1)

The integral representation reads as:

$$W_{\lambda ,\mu }\left(z\right)=\frac{1}{2\pi i}{\int }_{H{a}_{-}}{e}^{\sigma +z{\sigma }^{-\lambda }}\frac{d\sigma }{{\sigma }^{\mu }},\lambda >-1,\mu \in \mathbb{C},$$

(2)

where $H{a}_{-}$ denotes the Hankel path: this one is a loop which starts from $-\infty$ along the lower side of negative real axis, encircling it with a small circle the axes origin and ends at $-\infty$ along the upper side of the negative real axis.

$W_{\lambda ,\mu }\left(z\right)$ is then an entire function for all $\lambda \in (-1,+\infty )$. Originally, Wright assumed $\lambda \ge 0$ in connection with his investigations on the asymptotic theory of partition [4,5] and only in 1940 he considered $-1<\lambda <0$, [6]. We note that, in the Vol 3, Chapter 18 of the handbook of the Bateman Project [7], presumably for a misprint, the parameter $\lambda$ is restricted to be non-negative, whereas the Wright functions remained practically ignored in other handbooks. In 1993, Mainardi, being aware only of the Bateman handbook, proved that the Wright function is entire also for $-1<\lambda <0$ in his approaches to the time fractional diffusion equation that will be dealt with in the next section.

In view of the asymptotic representation in the complex domain and of the Laplace transform for positive argument $z=r>0$ (r can be the time variable t or the space variable x), the Wright functions are distinguished in first kind ($\lambda \ge 0$) and second kind ($-1<\lambda <0$) as outlined in the Appendix F of the book by Mainardi [8]. In particular, for the asymptotic behavior, we refer the interested reader to the two papers by Wong and Zhao [9,10], and to the surveys by Luchko and by Paris in the Handbook of Fractional Calculus and Applications, see, respectively, [11,12], and references therein.

We note that the Wright functions are an entire of order $1/(1+\lambda )$; hence, only the first kind functions ($\lambda \ge 0$) are of exponential order, whereas the second kind functions ($-1<\lambda <0$) are not of exponential order. The case $\lambda =0$ is trivial since $W_{0,\mu }\left(z\right)={\mathrm{e}}^z/\mathsf{\Gamma }\left(\mu \right).$ As a consequence of the difference in the orders, we must point out the different Laplace transforms proved e.g., in [8,13], see also the recent survey on Wright functions by Luchko [11]. We have:

• for the first kind, when $\lambda \ge 0$

  $$W_{\lambda ,\mu }(\pm r)÷\frac{1}{s}{E}_{\lambda ,\mu }\left(\pm \frac{1}{s}\right);$$

  (3)
• for the second kind, when $-1<\lambda <0$ and putting for convenience $\nu =-\lambda$ so $0<\nu <1$

  $$W_{-\nu ,\mu }(-r)÷E_{\nu ,\mu +\nu }\left(-s\right).$$

  (4)

Above, we have introduced the Mittag–Leffler function in two parameters $\alpha >0$, $\beta \in \mathbb{C}$ defined as its convergent series for all $z\in \mathbb{C}$

$$E_{\alpha ,\beta }\left(z\right):=\sum _{n=0}^{\infty }\frac{z^n}{\mathsf{\Gamma }(\alpha n+\beta )}.$$

(5)

For more details on the special functions of the Mittag–Leffler type, we refer the interested readers to the treatise by Gorenflo et al. [14], where, in the forthcoming 2nd edition, the Wright functions are also treated in some detail.

In particular, two Wright functions of the second kind, originally introduced by Mainardi and named $F_{\nu }\left(z\right)$ and $M_{\nu }\left(z\right)$ ($0<\nu <1$), are called auxiliary functions in virtue of their role in the time fractional diffusion equations considered in the next section. These functions, $F_{\nu }\left(z\right)$ and $M_{\nu }\left(z\right)$, are indeed special cases of the Wright function of the second kind $W_{\lambda ,\mu }\left(z\right)$ by setting, respectively, $\lambda =-\nu$ and $\mu =0$ or $\mu =1-\nu$. Hence, we have:

$$F_{\nu }\left(z\right):=W_{-\nu ,0}(-z),0<\nu <1,$$

(6)

and

$$M_{\nu }\left(z\right):=W_{-\nu ,1-\nu }(-z),0<\nu <1.$$

(7)

Those functions are interrelated through the following relation:

$$F_{\nu }\left(z\right)=\nu z{M}_{\nu }\left(z\right),$$

(8)

which reminds us of the second relation in (A9), seen for the standard diffusion equation.

The series representations of the auxiliary functions are derived from those of $W_{\lambda ,\mu }\left(z\right)$. Then:

$$F_{\nu }\left(z\right):=\sum _{n=1}^{\infty }\frac{{(-z)}^n}{n!\mathsf{\Gamma }(-\nu n)}=\frac{1}{\pi }\sum _{n=1}^{\infty }\frac{{(-z)}^{n-1}}{n!}\mathsf{\Gamma }(\nu n+1)sin\left(\pi \nu n\right)$$

(9)

and

$$M_{\nu }\left(z\right):=\sum _{n=0}^{\infty }\frac{{(-z)}^n}{n!\mathsf{\Gamma }[-\nu n+(1-\nu \left)\right]}=\frac{1}{\pi }\sum _{n=1}^{\infty }\frac{{(-z)}^{n-1}}{(n-1)!}\mathsf{\Gamma }\left(\nu n\right)sin\left(\pi \nu n\right),$$

(10)

where in both cases the reflection formula for the Gamma function (Equation (11)) it has been used among the first and the second step of Equations (9) and (10),

$$\mathsf{\Gamma }\left(\zeta \right)\mathsf{\Gamma }(1-\zeta )=\pi /sin\pi \zeta .$$

(11)

In addition, the integral representations of the auxiliary functions are derived from those of $W_{\lambda ,\mu }\left(z\right)$. Then:

$$F_{\nu }\left(z\right):=\frac{1}{2\pi i}{\int }_{H{a}_{-}}{e}^{\sigma -z{\sigma }^{\nu }}d\sigma ,z\in \mathbb{C},0<\nu <1$$

(12)

and

$$M_{\nu }\left(z\right):=\frac{1}{2\pi i}{\int }_{H{a}_{-}}{e}^{\sigma -z{\sigma }^{\nu }}\frac{d\sigma }{{\sigma }^{1-\nu }},z\in \mathbb{C},0<\nu <1.$$

(13)

Explicit expressions of $F_{\nu }\left(z\right)$ and $M_{\nu }\left(z\right)$ in terms of known functions are expected for some particular values of $\nu$ as shown and recalled by Mainardi in the first 1990s in a series of papers [15,16,17,18] that is,

$$M_{1/2}\left(z\right)=\frac{1}{\sqrt{\pi }}{e}^{-z^2/4},$$

(14)

$$M_{1/3}\left(z\right)=3^{2/3}\mathrm{Ai}(z/3^{1/3}).$$

(15)

Liemert and Klenie [19] have added the following expression for $\nu =2/3$

$$M_{2/3}\left(z\right)=3^{-2/3}\left[3^{1/3}z\mathrm{Ai}\left(z^2/3^{4/3}\right)-3{\mathrm{Ai}}^{\prime }\left(z^2/3^{4/3}\right)\right]{\mathrm{e}}^{-2z^3/27},$$

(16)

where Ai and ${\mathrm{Ai}}^{\prime }$ denote the Airy function and its first derivative. Furthermore, they have suggested in the positive real field ${\mathbb{R}}^{+}$ the following remarkably integral representation

$$M_{\nu }\left(x\right)=\frac{1}{\pi }\frac{x^{\nu /(1-\nu )}}{1-\nu }{\int }_0^{\pi }{C}_{\nu }\left(\varphi \right)\exp \left(-C_{\nu }\left(\varphi \right)\right)x^{1/(1-\nu )}d\varphi ,$$

(17)

where

$$C_{\nu }\left(\varphi \right)=\frac{sin(1-\nu )}{sin\varphi }{\left(\frac{sin\nu \varphi }{sin\varphi }\right)}^{\nu /(1-\nu )}$$

(18)

corresponding to Equation (7) of the article written by Saa and Venegeroles [20] .

The Wright function of both kinds and in particular the Mainardi auxiliary functions considerd in this paper turn out to be particular cases of more general transcendental functions as the Fox H functions, the Fox–Wright functions and the multi-index Mittag–Leffler functions. The relations with the classical Mittag–Leffler functions with two parameters have already been pointed out so; for more parameters, we refer the interested reader, e.g., to the papers by Kiryakova [21], Kilbas, Koroleva, Rogosin [22], and references therein.

We outline that for more Laplace transform pairs involving the Wright and the Mittag–Leffler functions the reader is referred to Ansari and Refahi Sheikhani [23] and to the tutorial survey by Mainardi [24].

3. The Wright Functions of the Second Kind and the Time-Fractional Diffusion Wave Equation

As we will see, the Wright functions of the second kind are relevant in the analysis of the Time-Fractional Diffusion-Wave Equation (TFDWE).

We find it convenient to show the plots of the M-Wright functions on a space symmetric interval of $\mathbb{R}$ in Figure 1 and Figure 2, corresponding to the cases $0\le \nu \le 1/2$ and $1/2\le \nu \le 1$, respectively.

From these figures, we recognize the non-negativity of the M-Wright function on $\mathbb{R}$ for $1/2\le \nu \le 1$ consistently with the analysis on distribution of zeros and asymptotics of Wright functions carried out by Luchko, see [11,25] and by Luchko and Kiryakova [26].

For this purpose, we introduce now the TFDWE as a generalization of the standard diffusion equation and we see how the two Mainardi auxiliary functions come into play. The TFDWE is thus obtained from the standard diffusion equation (or the D’Alembert wave equation) by replacing the first-order (or the second-order) time derivative by a fractional derivative (of order $0<\beta \le 2$) in the Caputo sense, obtaining the following Fractional PDE:

$$\frac{{\partial }^{\beta }u}{\partial t^{\beta }}=D\frac{{\partial }^{2}u}{\partial x^2}0<\beta \le 2,D>0,$$

(19)

where D is a positive constant whose dimensions are $L^2{T}^{-\beta }$ and $u=u(x,t;\beta )$ is the field variable, which is assumed again to be a causal function of time. The Caputo fractional derivative is recalled in the Appendix B so that in explicit form the TFDWE (19) splits in the following integro-differential equations:

$$\frac{1}{\mathsf{\Gamma }(1-\beta )}{\int }_0^t{(t-\tau )}^{-\beta }\left(\frac{\partial u}{\partial \tau }\right)d\tau =D\frac{{\partial }^{2}u}{\partial x^2},0<\beta \le 1;$$

(20)

$$\frac{1}{\mathsf{\Gamma }(2-\beta )}{\int }_0^t{(t-\tau )}^{1-\beta }\left(\frac{{\partial }^{2}u}{\partial {\tau }^2}\right)d\tau =D\frac{{\partial }^{2}u}{\partial x^2},1<\beta \le 2.$$

(21)

In view of our analysis, we find it convenient to put:

$$\nu =\frac{\beta }{2},0<\nu \le 1.$$

(22)

We can then formulate the basic problems for the Time Fractional Diffusion-Wave Equation using a correspondence with the two problems for the standard diffusion equation.

Denoting by $f\left(x\right)$ and $g\left(t\right)$ two given, sufficiently well-behaved functions, we define:

(a)

Cauchy problem

$$\left\{\begin{array}{cc}u(x,0^{+};\nu )=f\left(x\right),-\infty <x<+\infty ;\hfill & \hfill \\ u(\pm \infty ,t;\nu )=0,t>0\hfill & \hfill \end{array}\right.$$

(23)

(b)

Signalling problem

$$\left\{\begin{array}{c}u(x,0^{+};\nu )=0,0\le x<+\infty ;\hfill \\ u(0^{+},t;\nu )=g\left(t\right),u(+\infty ,t;\nu )=0,t>0\hfill & \hfill \end{array}\right.$$

(24)

If $1/2<\nu \le 1$ corresponding to $1<\beta \le 2$, we must consider also the initial value of the first time derivative of the field variable $u_t(x,0^{+};\nu )$, since, in this case, Equation (19) turns out to be akin to the wave equation and consequently two linear independent solutions are to be determined. However, to ensure the continuous dependence of the solutions to our basic problems on the parameter $\nu$ in the transition from $\nu ={(1/2)}^{-}$ to $\nu ={(1/2)}^{+}$, we agree to assume $u_t(x,0^{+};\nu )=0$.

For the Cauchy and Signalling problems, following the approaches by Mainardi, see, e.g., [15] and related papers, we introduce now the Green functions ${\mathcal{G}}_c(x,t;\nu )$ and ${\mathcal{G}}_s(x,t;\nu )$ that for both problems can be determined by the $LT$ technique, so extending the results known from the ordinary diffusion equation. We recall that the Green functions are also referred to as the fundamental solutions, corresponding respectively to $f\left(x\right)=\delta \left(x\right)$ and $g\left(t\right)=\delta \left(t\right)$ with $\delta (·)$ is the Dirac delta generalized function

The expressions for the Laplace Transforms of the two Green’s functions are:

$${\tilde{\mathcal{G}}}_c(x,s;\nu )=\frac{1}{2\sqrt{D}{s}^{1-\nu }}{\mathrm{e}}^{(-|x|/\sqrt{D})s^{\nu }}$$

(25)

and

$${\tilde{\mathcal{G}}}_s(x,s;\nu )={\mathrm{e}}^{-(x/\sqrt{D})s^{\nu }}$$

(26)

Now, we can easily recognize the following relation:

$$\frac{d}{ds}{\tilde{\mathcal{G}}}_s=-2\nu x{\tilde{\mathcal{G}}}_c,x>0$$

(27)

which implies for the original Green functions the following reciprocity relation for $x>0$`and $t>0$ and $0<\nu <1$:

$$2\nu x{\mathcal{G}}_c(x,t;\nu )=t{\mathcal{G}}_s(x,t;\nu )=F_{\nu }\left(z\right)=\nu z{M}_{\nu }\left(z\right)z=\frac{x}{\sqrt{D}{t}^{\nu }}$$

(28)

where z is the similarity variable and $F_{\nu }\left(z\right)$ and $M_{\nu }\left(z\right)$ are the Mainardi auxilary functions introduced in the previous section. Indeed, Equation (28) is the generalization of Equation (A8) that we have seen for the standard diffusion equation due to the introduction of the time fractional derivative of order $\nu$.

Then, the two Green functions of the Cauchy and Signalling problems turn out to be expressed in terms of the two auxiliary functions as follows.

For the Cauchy problem, we have

$${\mathcal{G}}_C(x,t;\nu )=\frac{t^{-\nu }}{2\sqrt{D}}{M}_{\nu }\left(\frac{\left|x\right|}{\sqrt{D}{t}^{\nu }}\right)-\infty <x<+\infty t\ge 0$$

(29)

that generalizes Equation (A5).

For the Signalling problem, we have:

$${\mathcal{G}}_S(x,t;\nu )=\frac{\nu x{t}^{-\nu -1}}{\sqrt{D}}{M}_{\nu }\left(\frac{x}{\sqrt{D}{t}^{\nu }}\right)x\ge 0,t\ge 0$$

(30)

that generalizes Equation (A7).

3.1. Complements to the Time-Fractional Diffusion-Wave Equations

The use of the Wright functions of the second kind in time fractional diffusion-wave equations has appeared in several papers for a variety of different purposes, see, e.g., Bazhlekova [27], D’Ovidio [28], Gorenflo, Luchko and Mainardi [29], Mentrelli and Pagnini [30], Mosley and Ansari [31], Pagnini [32], Povstenko [33], and references therein.

The boundary value problems dealt with previously can be considered with a source data function $f\left(x\right)$ and $g\left(t\right)$ different from the Dirac generalized functions, in particular with box-type functions as it has been carried out recently by us, see [34].

An interesting generalization of the TFDWE is obtained by considering time-fractional derivatives of distributed order. In this respect, we cite, e.g., the papers by Kochubei [35], Li, Luchko and Yamamoto [36], Mainardi, Pagnini and Gorenflo [37], and Mainardi et. al [38].

The TFDWE can also be generalized in 2D and 3D space dimensions. so consequently the Wright functions play again a fundamental role. However, we prefer to refer the interested reader to the literature, in particular to the papers by Luchko and collaborators [11,25,39,40,41,42,43], by Hanyga [44] and to the recent analysis by Kemppainen [45]. All of them are originated in some way from the seminal paper by Schneider and Wyss [46]. In some of these papers, the authors have considered also fractional differentiation both in time and in space, so that they have generalized to more than one dimension the former analysis by Mainardi, Luchko, and Pagnini [47] on the space-time fractional diffusion-wave equations.

4. The $\mathit{M}$-Wright Functions in Probability Theory and the Stable Distributions

We recognize that the Wright M-function with support in ${\mathbb{R}}^{+}$ can be interpreted as probability density function ($pdf$) because it is non negative and also it satisfies the normalization condition:

$${\int }_0^{\infty }{M}_{\nu }\left(x\right)dx=1.$$

(31)

We now provide more details on these densities in the framework of the theory of probability.

Theorem 1.

Let $M_{\nu }\left(x\right)$ be the M-Wright function in ${\mathbb{R}}^{+}$, $0\le \nu <1$ and $\delta >-1$. Then, the (finite)absolute momentsof order δ are given by:

$${\int }_0^{\infty }{x}^{\delta }{M}_{\nu }\left(x\right)dx=\frac{\mathsf{\Gamma }(\delta +1)}{\mathsf{\Gamma }(\nu \delta +1)}.$$

(32)

Proof. 

The proof is based on the integral representation of the M-Wright function:

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{x}^{\delta }{M}_{\nu }\left(x\right)dx& ={\int }_0^{\infty }{x}^{\delta }\left[\frac{1}{2\pi i}{\int }_{H{a}_{-}}{e}^{\sigma -x{\sigma }^{\nu }}\frac{d\sigma }{{\sigma }^{1-\nu }}\right]dx\hfill \\ \hfill & =\frac{1}{2\pi i}{\int }_{H{a}_{-}}{e}^{\sigma }\left[{\int }_0^{\infty }{e}^{-x{\sigma }^{\nu }}{x}^{\delta }dx\right]\frac{d\sigma }{{\sigma }^{1-\nu }}\hfill \\ \hfill & =\frac{\mathsf{\Gamma }(\delta +1)}{2\pi i}{\int }_{H{a}_{-}}\frac{e^{\sigma }}{{\sigma }^{\nu \delta +1}}d\sigma =\frac{\mathsf{\Gamma }(\delta +1)}{\mathsf{\Gamma }(\nu \delta +1)}\hfill \end{array}$$

(33)

□

The exchange between two integrals and the following identity contributed to the final result for Equation (33):

$${\int }_0^{\infty }{\mathrm{e}}^{-x{\sigma }^{\nu }}{x}^{\delta }dx=\frac{\mathsf{\Gamma }(\delta +1)}{{\left({\sigma }^{\nu }\right)}^{\delta +1}}.$$

(34)

In particular, for $\delta =n\in \mathbb{N}$, the above formula provides the moments of integer order. Indeed, recalling the Mittag–Leffler function introduced in Equation (5) with $\alpha =\nu$ and $\beta =1$:

$$E_{\nu }\left(z\right):=\sum _{n=0}^{\infty }\frac{z^n}{\mathsf{\Gamma }(\nu n+1)},\nu >0,z\in \mathbb{C},$$

(35)

the moments of integer order can also be computed from the Laplace transform pair

$$M_{\nu }\left(x\right)÷E_{\nu }(-s)$$

(36)

proved in the Appendix F of [8] as follows:

$${\int }_0^{+\infty }{x}^n{M}_{\nu }\left(x\right)dx=\underset{s\to 0}{lim}{(-1)}^n\frac{d^n}{d{s}^n}{E}_{\nu }(-s)=\frac{\mathsf{\Gamma }(n+1)}{\mathsf{\Gamma }(\nu n+1)}.$$

(37)

4.1. The Auxiliary Functions versus Extremal Stable Densities

We find it worthwhile to recall the relations between the Mainardi auxiliary functions and the extremal Lévy stable densities as proven in the 1997 paper by Mainardi and Tomirotti [48]. For readers’ convenience, we refer to Appendix C for an essential account of the general Lévy stable distributions in probability. Indeed, from a comparison between the series expansions of stable densities in (A41) and (A42) and of the auxiliary functions in Equations (9) and (10), we recognize that the auxiliary functions are related to the extremal stable densities as follows:

$$L_{\alpha }^{-\alpha }\left(x\right)=\frac{1}{x}{F}_{\alpha }\left(x^{-\alpha }\right)=\frac{\alpha }{x^{\alpha +1}}{M}_{\alpha }\left(x^{-\alpha }\right)0<\alpha <1x\ge 0$$

(38)

$$L_{\alpha }^{\alpha -2}\left(x\right)=\frac{1}{x}{F}_{1/\alpha }\left(x\right)=\frac{1}{\alpha }{M}_{1/\alpha }\left(x\right)1<\alpha \le 2-\infty <x<+\infty .$$

(39)

In the above equations, for $\alpha =1$, the skewness parameter turns out to be $\theta =-1$, so we get the singular limit

$$L_1^{-1}\left(x\right)=M_1\left(x\right)=\delta (x-1).$$

(40)

Hereafter, we show in Figure 3 and Figure 4 the plots the extremal stable densities according to their expressions in terms of the M-Wright functions, see Equations (38) and (39) for $\alpha =1/2$ and $\alpha =3/2$, respectively.

We recognize that the above plots are consistent with the corresponding ones shown by Mainardi et al. [47] for the stable pdf’s derived as fundamental solutions of a suitable space-fractional diffusion equation.

4.2. The Symmetric M-Wright Function

We easily recognize that extending the function $M_{\nu }\left(x\right)$ in a symmetric way to all of $\mathbb{R}$ (that is putting $x=\left|x\right|$) and dividing by 2 we have a symmetric $pdf$ with support in all of $\mathbb{R}$.

As the parameter $\nu$ changes between 0 and 1, the pdf goes from the Laplace pdf to two half discrete delta pdfs passing for $\nu =1/2$ through the Gaussian pdf.

To develop a visual intuition, also in view of the subsequent applications, we show n Figure 5 and Figure 6 the plots of the symmetric M-Wright function on the real axis at $t=1$ for some rational values of the parameter $\nu \in [0,1]$

The readers are invited to look the YouTube video by Consiglio whose title is “Simulation of the M-Wright function”, in which the author shows the evolution of this function as the parameter $\nu$ changes between 0 and 0.85 in a finite interval of $\mathbb{R}$ centered in $x=0$.

Theorem 2.

Let $M_{\nu }\left(\right|x\left|\right)$ be the symmetric M-Wright function pdf. Then, its characteristic function is:

$$\begin{array}{c}\hfill \mathcal{F}\left[\frac{1}{2}{M}_{\nu }\left(\right|x\left|\right)\right]=E_{2\nu }(-{\kappa }^2)\end{array}$$

(41)

Proof. 

The proof is based on the series development of the cosine function and on Equation (33):

$$\begin{array}{cc}\hfill \mathcal{F}\left[\frac{1}{2}{M}_{\nu }\left(\right|x\left|\right)\right]& :=\frac{1}{2}{\int }_{-\infty }^{+\infty }{e}^{+i\kappa x}{M}_{\nu }\left(\right|x\left|\right)dx\hfill \\ \hfill & ={\int }_0^{\infty }cos\left(\kappa x\right)M_{\nu }\left(x\right)dx\hfill \\ \hfill & =\sum _{n=0}^{\infty }{(-1)}^n\frac{{\kappa }^{2n}}{\left(2n\right)!}{\int }_0^{\infty }{x}^{2n}{M}_{\nu }\left(x\right)dx\hfill \\ \hfill & =\sum _{n=0}^{\infty }{(-1)}^n\frac{{\kappa }^{2n}}{\mathsf{\Gamma }(2\nu n+1)}=E_{2\nu }(-{\kappa }^2)\hfill \end{array}$$

(42)

□

4.3. The Wright $\mathbb{M}$-Function in Two Variables

In view of time-fractional diffusion processes related to time-fractional diffusion equations, it is worthwhile to introduce the function in two variables

$${\mathbb{M}}_{\nu }(x,t):=t^{-\nu }{M}_{\nu }\left(x{t}^{-\nu }\right)0<\nu <1x,t\in {\mathbb{R}}^{+}$$

(43)

which defines a spatial probability density in x evolving in time t with self-similarity exponent $H=\nu$. Of course, for $x\in \mathbb{R}$, we have to consider the symmetric version of the M-Wright function. Hereafter, we provide a list of the main properties of this function, which can be derived from the Laplace and Fourier transforms for the corresponding Wright M-function in one variable.

From Equations (39) and (43), we derive the Laplace transform of ${\mathbb{M}}_{\nu }(x,t)$ with respect to $t\in {\mathbb{R}}^{+}$,

$$\mathcal{L}\left\{{\mathbb{M}}_{\nu }(x,t);t\to s\right\}=s^{\nu -1}{\mathrm{e}}^{{\displaystyle -x{s}^{\nu }}}.$$

(44)

From Equation (18), we derive the Laplace transform of ${\mathbb{M}}_{\nu }(x,t)$ with respect to $x\in {\mathbb{R}}^{+}$,

$$\mathcal{L}\left\{{\mathbb{M}}_{\nu }(x,t);x\to s\right\}=E_{\nu }\left(-s{t}^{\nu }\right).$$

(45)

From Equation (55), we derive the Fourier transform of ${\mathbb{M}}_{\nu }\left(\right|x|,t)$ with respect to $x\in \mathbb{R}$,

$$\mathcal{F}\left\{{\mathbb{M}}_{\nu }\left(\right|x|,t);x\to \kappa \right\}=2E_{2\nu }\left(-{\kappa }^2{t}^{\nu }\right).$$

(46)

Using the Mellin transforms, Mainardi et al. [49] derived the following interesting integral formula of composition,

$${\mathbb{M}}_{\nu }(x,t)={\int }_0^{\infty }{\mathbb{M}}_{\lambda }(x,\tau ){\mathbb{M}}_{\mu }(\tau ,t)d\tau \nu =\lambda \mu .$$

(47)

Special cases of the Wright $\mathbb{M}$-function are simply derived for $\nu =1/2$ and $\nu =1/3$ from the corresponding ones in the complex domain, see Equations (28) and (29). We devote particular attention to the case $\nu =1/2$ for which we get the Gaussian density in $\mathbb{R}$,

$${\mathbb{M}}_{1/2}\left(\right|x|,t)=\frac{1}{2\sqrt{\pi }{t}^{1/2}}{\mathrm{e}}^{{\displaystyle -x^2/\left(4t\right)}}.$$

(48)

For the limiting case $\nu =1$, we obtain

$${\mathbb{M}}_1\left(\right|x|,t)=\frac{1}{2}\left[\delta (x-t)+\delta (x+t)\right].$$

(49)

We conclude this section pointing out that the M-Wright functions have been applied by several authors in the theory of probability and stochastic processes, see, e.g., Beghin and Orsingher [50], Cahoy [51,52], Garra, Orsingher and Polito [53], Le Chen [54], Consiglio, Luchko and Mainardi [55], Gorenflo and Mainardi [56], Mainardi, Mura and Pagnini [57], Pagnini [58], Scalas and Viles [59], and references therein. Furthermore, these functions have been found in the first passage problem for Lévy flights dealt by the group of Prof. Metzler, see e.g., [60,61].

5. The Four Sisters

In this section, we show how some Wright functions of the second kind can provide an interesting generalization of the three sisters discussed in Appendix A. The starting point is a (not well- known) paper published in 1970 by Stankovic [62], where (in our notation) the following Laplace transform pair is proved rigorously:

$$t^{\mu -1}{W}_{-\nu ,\mu }(x,t)÷s^{-\mu }{\mathrm{e}}^{-x{s}^{\nu }}0<\nu <1\mu \ge 0$$

(50)

where x and t are positive. We note that the Stankovic formula can be derived in a formal way by developing the exponential function in positive power of s and inverting term by term as described in the Appendix F of the book by Mainardi [8].

We recognize that the Laplace Transforms of the Three Sisters functions $\tilde{\varphi }(x,s)$, $\tilde{\psi }(x,s)$ and $\tilde{\chi }(x,s)$ are particular cases of the Equation (50) for $\nu =1/2$ that is of

$$t^{\mu -1}{W}_{-1/2,\mu }(x,t)÷s^{-\mu }{\mathrm{e}}^{-x\sqrt{s}},$$

(51)

according to the following scheme:

$$\tilde{\varphi }(x,s)\mathrm{with}\mu =1;\tilde{\psi }(x,s)\mathrm{with}\mu =0;\tilde{\chi }(x,s)\mathrm{with} \mu =1/2.$$

If $\nu$ is no longer restricted to $\nu =1/2$, we define Four Sisters functions as follows:

$$\begin{array}{cc}\hfill \mu =0,{\mathrm{e}}^{-x{s}^{\nu }}& ÷t^{-1}{W}_{-\nu ,0}(-x{t}^{-\nu }),\hfill \\ \hfill \mu =1-\nu ,\frac{{\mathrm{e}}^{-x{s}^{\nu }}}{s^{1-\nu }}& ÷t^{-\nu }{W}_{-\nu ,1-\nu }(-x{t}^{-\nu }),\hfill \\ \hfill \mu =\nu ,\frac{{\mathrm{e}}^{-x{s}^{\nu }}}{s^{\nu }}& ÷t^{\nu -1}{W}_{-\nu ,\nu }(-x{t}^{-\nu }),\hfill \\ \hfill \mu =1,\frac{{\mathrm{e}}^{-x{s}^{\nu }}}{s}& ÷W_{-\nu ,1}(-x{t}^{-\nu }).\hfill \end{array}$$

(52)

Hereafter, in Figure 7, Figure 8 and Figure 9, we show some plots of these functions, both in the t and in the x domain for some values of $\nu$ ($\nu =1/4,1/2,3/4$).

Note that for $\nu =1/2$ we only find three functions, that is the Three Sisters functions of Appendix A.

6. Conclusions

In our survey on the Wright functions, we have distinguished two kinds, pointing out the particular class of the second kind. Indeed, these functions have been shown to play key roles in several processes governed by non-Gaussian processes, including sub-diffusion, transition to wave propagation, Lévy stable distributions. Furthermore, we have devoted our attention to four functions of this class that we agree to called the Four Sisters functions. All these items justify the relevance of the Wright functions of the second kind in Mathematical Physics.