Finite Representations of theWright Function

1. Introduction

The Wright function was introduced in two seminal publications by the British mathematician Sir E.M. Wright discussing the theory of partitions of numbers [1,2]. The function received renewed interest from the mathematical community when it was demonstrated that the space-time fractional diffusion equation with the temporal Caputo derivative can be solved in terms of Wright functions [3]. It was also discovered that the Wright function provides a unified treatment of several classes of special functions, notably the Bessel functions, the probability integral erf the Airy Ai, and the Whittaker function, among others. The Wright function was originally defined by the infinite series [1]:

$$W\left(a,b|z\right):=\sum _{k=0}^{\infty }\frac{z^k}{k!\Gamma (ak+b)},z\in \mathbb{C},$$

under the conditions $b\in \mathbb{C}$ and $a>-1$, where $\Gamma$ denotes the Euler’s Gamma function. Later works on the function include the articles of Gorenflo, Luchko and Mainardi [4] and Luckko [5] among some others. Based on the sign of its first parameter, later, Mainardi classified the function into two types: the Wright function of the 1^st type if $a\ge 0$ and the Wright function of 2^nd type for $-1<a<0$ [6,7]. This function fits into the more general theory of the Fox-Wright (FW) functions as will be discussed in Sec. 3.

The Wright function is closely related to the theory of the generalized hypergeometric (GHG) functions. Notably, for rational parameter values the Wright function can be represented as a finite sum of GHG functions. The link comes directly through the theory of the Euler’s Gamma function. Formulas for the Wright function representation of the 1^st type have been published in [4,8] and have been derived through its representation as a Meijer G function. Recently, Apelblat and Gonzales-Santander have tabulated representations in terms of GHG functions for many parameter combinations [9].

The contribution of the present article is twofold. On the first place, it extends the results of the above authors [9] for the cases wherever $a<0$ and $b>1$ and also demonstrates how the domain of the first parameter can be extended into the negative integers under certain conditions by explicitly constructing polynomial representations of the function. These representations allow us to distinguish a Wright function of 3^rd type (Sec. 4). Some of the present results have been presented in a preliminary form at the 2023 International Conference on Fractional Differentiation and Its Applications ICFDA 2023 [10]. On the second place, the article exhibits the link with the Mittag-Leffler function, which also has wide applications in fractional calculus. It is demonstrated that the theory of the Wright function is very rich and can produce many potentially useful integral identities. In a similar way, the domain of the Mittag-Leffler function can be analytically continued into negative integral values of its first parameter and integer values of its second parameter.

2. Some Applications of the Wright Function

Recent surveys about Wright function applications can be found in [7] and [11]. What makes the function useful for applications in calculus is the fact that it is closed under differentiation

$$\frac{d}{dz}W\left(\left.a,b\right|z\right)=W\left(\left.a,a+b\right|z\right)$$

(1)

which allows one to write

$$W\left(\left.a,b\right|z\right)={\int }_0^{z}W\left(\left.a,a+b\right|z\right)+\frac{1}{\Gamma \left(b\right)}$$

(2)

and

$$\int W\left(\left.a,b\right|z\right)dz=W\left(\left.a,b-a\right|z\right)+C$$

(3)

The Wright function arises in the theory of the space-time fractional diffusion equation (FDE) with the temporal Caputo derivative [3]. We recall that the Caputo’s fractional derivative of order $\beta >0$ is defined for $\beta \notin \mathbb{N}$ as the differ-integral

$${\mathcal{D}}_t^{\beta }f\left(t\right):=\frac{1}{\Gamma (m-\beta )}{\int }_0^t\frac{f^{\left(m\right)}\left(u\right)du}{{(t-u)}^{\beta +1-m}}$$

(4)

where $m=\lfloor \beta \rfloor$. The fractional differential equation in the Caputo sense with variable coefficients

$${\mathcal{D}}_t^b\left[t^b{f}^{\prime }\left(t\right)\right]=a{t}^{b-1}f\left(t\right)$$

(5)

admits for a solution $f\left(t\right)=W\left(\left.a,b\right|t^a\right)$ [12].

3. The Wright Function as a Simple Representative of the Fox-Wright Function Family

The generalized hypergeometric functions are defined by the infinite hypergeometric (HG) series

$$\begin{array}{cc}{}_{p}F_q(a_1,\dots ,a_p;b_1,\dots ,b_q,x):=\hfill & \\ & \hfill \sum _{m=0}^{\infty }\frac{x^m}{\Gamma (m+1)}\prod _{k=1}^p\frac{\Gamma (a_k+m)}{\Gamma \left(a_k\right)}\prod _{k=1}^q\frac{\Gamma \left(b_k\right)}{\Gamma (b_k+m)}=\sum _{r=0}^{\infty }\frac{x^r}{r!}\frac{{\prod }_{j=0}^{p-1}{\left(a_j\right)}_r}{{\prod }_{j=0}^{q-1}{\left(b_j\right)}_r}\end{array}$$

(6)

where ${\left(a\right)}_r$ and ${\left(b\right)}_r$ will denote rising factorials and ${\left(a\right)}_0=1$, which assumes the normalization ${}_{p}F_q\left(\sim ;\sim |0\right)=1$. By convention, equal parameters in the numerator and denominator will cancel out. Unless stated otherwise it will be always assumed that the infinite series converge in some domain $x\in \mathbb{R}$.

The defining property fo HG series is that the coefficients are rational functions of the index variable (i.e. k). In the present article we will use the parametric notation similar to the one adopted by Oldham and Spanier [13].

$${}_{p}F_q(a_1,\dots ,a_p;b_1,\dots ,b_q,x)\equiv \left[\begin{array}{ccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}x\hfill \\ \end{array}\end{array}\right]$$

The FW functions are further generalizations of the hypergeometric (HG) functions. It can be defined by the infinite series

$$\begin{array}{cc}{}_p\overline{\Psi }_q\left(x\right)\equiv \overline{\Psi }\left[\begin{array}{ccc}\begin{array}{c}(A_1,a_1)\dots ,(A_p,a_p)\\ (B_1,b_1)\dots ,(B_q,b_q)\end{array}& \mid & \begin{array}{c}x\hfill \\ \end{array}\end{array}\right]:=\hfill & \\ & \hfill \sum _{m=0}^{\infty }\frac{x^m}{\Gamma (m+1)}\prod _{k=1}^p\frac{\Gamma (a_{k}m+A_k)}{\Gamma \left(A_k\right)}\prod _{k=1}^q\frac{\Gamma \left(B_k\right)}{\Gamma (b_{k}m+B_k)}\end{array}$$

whenever it converges.

At this point the following extended notation is introduced under the convention

$${}_{p+1}\overline{\Psi }_q\left(z\right)\equiv \left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}(A,a)\\ -\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right],{}_{p+1}\overline{\Psi }_q\left(0\right)=1,$$

In this notation, the hypergeometric parameters of the function are written first while the composite parameters are left second. The right parameters result in factors of the form

$$\frac{\Gamma (ka+A)}{\Gamma \left(A\right)}$$

or their reciprocals, respectively, while the left parameters result is Pochhammer multipliers (i.e. $A\in \mathbb{N}$). The non-simplified parameters follow the usual convention established in literature. The order in the parametric convention for the arguments of the Gamma function follows the usual convention.

The following simplifying convention will be used further:

$$\left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots \\ b_1,\dots \end{array}& \mid & \begin{array}{c}-\\ -\end{array}& \mid & \begin{array}{c}x\hfill \\ \end{array}\end{array}\right]\equiv \left[\begin{array}{ccc}\begin{array}{c}{a}_1,\dots \\ b_1,\dots \end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]$$

(7)

and

$$\left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}(A,1)\\ -\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]\equiv \left[\begin{array}{ccc}\begin{array}{c}{a}_1,\dots ,a_p,A\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]$$

(8)

This example shows different ways to write a hypergeometric function. Under this notation

$$W\left(\left.a,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}\left[\begin{array}{ccccc}\begin{array}{c}-\\ -\end{array}& \mid & \begin{array}{c}-\\ (b,a)\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]$$

(9)

In this way one could appreciate that the Wright function is the simplest member of the class of the Fox-Wright functions. Other examples are the Bessel J function:

$$J_{\nu }\left(z\right)=\frac{1}{\Gamma (\nu +1)}{\left(\frac{z}{2}\right)}^{\nu }\left[\begin{array}{ccc}\begin{array}{c}-\\ \nu +1\end{array}& \mid & \begin{array}{c}-\frac{z^2}{4}\hfill \\ \end{array}\end{array}\right]$$

The Struve H function:

$$H_{\nu }\left(z\right)=\frac{1}{\Gamma (\nu +3/2)\Gamma (3/2)}{\left(\frac{z}{2}\right)}^{\nu +1}\left[\begin{array}{ccc}\begin{array}{c}1\\ 3/2,\nu +3/2\end{array}& \mid & \begin{array}{c}-\frac{z^2}{4}\hfill \\ \end{array}\end{array}\right]$$

Furthermore, the following integral representation can be derived (see for example [14]):

$$\begin{array}{c}\hfill \left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}\dots \\ (B,b)\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]=\frac{\Gamma \left(B\right)}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\tau }}{{\tau }^B}\left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}\dots \\ \dots \end{array}& \mid & \begin{array}{c}\frac{z}{{\tau }^b}\hfill \\ \end{array}\end{array}\right]d\tau \end{array}$$

(10)

where $H{a}^{-}$ denotes the Hankel contour, which surrounds all poles of the GHG function from the left. Applied to the Wright function, where $B\mapsto a;b\mapsto a$ this gives the integral

$$W\left(\left.a,b\right|z\right)=\frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\xi +z{\xi }^{-a}}}{{\xi }^b}d\xi ,z\in \mathbb{C}$$

(11)

along a Hankel contour, which surrounds the negative real semi-axis and the pole at the origin. Said contour can be deformed in a extreme was as depicted in Figure 1. This contour consists of the rays $AB$ and $DE$ as well as the arc $BCD$. For integral values of b and a the path of integration closes around the origin O so that the rays collapse and can be used to extend the domain of the function into the negative integer parameters.

4. Polynomial Reduction

In particular, let us consider the case when a is a negative integer and denote it by $-n$. Trivially, if b is a negative integer, say $b=-m$, then the above integral vanishes and $W\left(\left.-n,-n\right|z\right)=0$.

In contrast, if $n=-a$ and $b=m,m$, such that $n\in \mathbb{N}$ then

$$W\left(\left.-n,m\right|z\right)=Res\left(ker\left(\xi \right),\xi =0\right)=\frac{1}{\Gamma \left(m\right)}\left\{{\left(\frac{d}{d\xi }\right)}^{m-1}{e}^{\xi +z{\xi }^n}\right\}{|}_{\xi =0}$$

Therefore, we can conclude that $W\left(\left.-n,m\right|z\right)$ is a polynomial in z. This is a novel result, which was not anticipated by Wright and allows for the extension of the domain of the parameters of the function. This polynomial can be computed explicitly by application of Faá di Bruno’s formula using the complete exponential Bell polynomials. For the natural numbers n and m:

$$W\left(\left.-n,m\right|z\right)=\frac{1}{\Gamma \left(m\right)}{B}_{m-1}\left(g^{\prime }\left(0\right),\dots ,g^{(m-1)}\left(0\right)\right)$$

(12)

where $g\left(\xi \right)=\xi +z{\xi }^n$ is the exponent of the kernel and it can be computed by the determinant

$$\begin{array}{c}\hfill B_m\left(g^{\prime }\left(0\right),\dots ,g^{\left(m\right)}\left(0\right)\right)=\left|\begin{array}{cccc}\left(\genfrac{}{}{0pt}{}{m-1}{0}\right)g^{\prime }\left(0\right)\hfill & \left(\genfrac{}{}{0pt}{}{m-2}{1}\right)g^{\prime \prime }\left(0\right)\hfill & \dots & \hfill \left(\genfrac{}{}{0pt}{}{m-1}{m-1}\right)g^{\left(m\right)}\left(0\right)\\ -1\hfill & \left(\genfrac{}{}{0pt}{}{m-2}{0}\right)g^{\prime }\left(0\right)\hfill & \dots & \hfill \left(\genfrac{}{}{0pt}{}{m-2}{m-2}\right)g^{(m-1)}\left(0\right)\\ 0\hfill & -1\hfill & \dots & \hfill \left(\genfrac{}{}{0pt}{}{m-3}{m-3}\right)g^{(m-3)}\left(0\right)\\ & & \dots \hfill & \\ 0\hfill & \dots \hfill & -1& \hfill \left(\genfrac{}{}{0pt}{}{0}{0}\right)g^{\prime }\left(0\right)\end{array}\right|\end{array}$$

Remark 1.

It should noted that the resulting matrix is a band matrix since already $g^{″}\left(0\right)=0$. For example, for $n=3$, $m=8$ we have

$$B_7(\dots )=\left|\begin{array}{ccccccc}1& 0& 90z& 0& 0& 0& 0\\ -1& 1& 0& 60z& 0& 0& 0\\ 0& -1& 1& 0& 36z& 0& 0\\ 0& 0& -1& 1& 0& 18z& 0\\ 0& 0& 0& -1& 1& 0& 6z\\ 0& 0& 0& 0& -1& 1& 0\\ 0& 0& 0& 0& 0& -1& 1\end{array}\right|$$

The polynomial reduction formulas allow us the claim that Mainardi’s classification can be extended to add also Wright functions of the third type, that is whenever $a,b\in {\mathbb{Z}}^{-}$.

5. Finite Hypergeometric Representations

Wherever the a parameter is rational the Wright function can be represented by a finite sum of hypergeometric functions. For positive and rational a one could obtain the representation in terms of ${}_{0}F_{m+n-1}$ GHG functions [9]:

Theorem 1

(First HG Representation). Suppose that $a=n/m>0$, where n and m are co-prime and $b\ne 0$. Then $W\left(\left.n/m,b\right|z\right)$ admits the finite representation

$$W\left(\left.n/m,b\right|z\right)=\sum _{r=0}^{m-1}\frac{z^r}{r!\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b},\overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{z^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right],$$

(13)

where $\overrightarrow{b}$ has n components and $\overrightarrow{c}$ has m components given by

$$b_j=r/m+(b+j)/n,c_j=(r+1+j)/m,$$

respectively.

The proof follows [8] and is given as a staring point for the proof of the Second Representation Theorem.

Proof. 

Starting from $a=n/m$

$$\begin{array}{c}\hfill W\left(\left.a,b\right|z\right)=\sum _{k=0}^{\infty }\frac{z^k}{k!\Gamma (ak+b)}=\sum _{q=0}^{m-1}\sum _{p\ge q/m}^{\infty }\frac{z^{mp-q}}{\Gamma (mp-q+1)\Gamma \left(a\right(mp-q)+b)}\end{array}$$

since the integer k can be partitioned as $k=mp-q$, where $q=0,\dots m-1$. After some algebra we obtain

$$W\left(\left.n/m,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}+\sum _{r=1}^m{z}^r\sum _{p=0}^{\infty }\frac{z^{mp}}{\Gamma (ap+ra+b)\Gamma (mp+r+1)}.$$

Observe that for $p=0$ the inner series evaluates to

$$C_r=\Gamma (ra+b)\Gamma (r+1),$$

which serves as its normalization factor. Therefore, the series transforms as

$$\begin{array}{c}\hfill W\left(\left.n/m,b\right|z\right)=\sum _{r=0}^m\frac{z^r}{C_r}·\sum _{p=0}^{\infty }\frac{\Gamma (ra+b)\Gamma (r+1)}{\Gamma \left(n(p+r/m+b/n)\right)}\frac{z^{mp}}{\Gamma \left(m(p+(r+1)/m)\right)}\end{array}$$

(14)

Further, use Prop. A1 to obtain

$$\frac{\Gamma \left(n(p+r/m)+b\right)}{\Gamma (nr/m+b)}=n^{np}\prod _{j=0}^{n-1}\underset{b_j}{\underbrace{{\left(\frac{r}{m}+\frac{b}{n}+\frac{j}{n}\right)}_p}}$$

(15)

From where we read off the component

$$b_0=\frac{r}{m}+\frac{b}{n}$$

with an increment $1/n$.

$$\frac{\Gamma \left(mp+r+1\right)}{\Gamma (r+1)}=m^{mp}\prod _{j=0}^{m-1}\underset{c_j}{\underbrace{{\left(\frac{r+1}{m}+\frac{j}{m}\right)}_p}}$$

(16)

From where we read off the component

$$c_0=\frac{r+1}{m}$$

with an increment $1/m$. □

Observe that $r=m-1$ results in $c_1=1$ therefore, the GHG functions reduce to ${}_{0}F_{m+n-1}$. The formula for a negative rational $a<0$ needs some more work. Suppose first that $b<1$. Let

$$\begin{array}{c}\hfill W\left(\left.-n/m,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}+\sum _{r=1}^m\frac{z^r}{C_r}\sum _{p=0}^{\infty }\frac{C_r}{\Gamma \left(-np-rn/m+b\right)}\frac{z^{mp}}{\Gamma \left(mp+r+1\right)}\end{array}$$

First we use the Gamma reflection formula to obtain

$$\frac{1}{\Gamma \left(-np-rn/m+b\right)}=\frac{{\left(-1\right)}^{np}\Gamma \left(\frac{nr}{m}+np-b+1\right)}{\Gamma \left(b-\frac{nr}{m}\right)\Gamma \left(\frac{nr}{m}-b+1\right)}$$

(17)

Therefore,

$$\begin{array}{c}W\left(\left.-n/m,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}+\hfill \\ \sum _{r=1}^m\frac{z^r}{C_r}\sum _{p=0}^{\infty }\frac{{(-1)}^{np}{C}_r\Gamma \left(\frac{nr}{m}+np-b+1\right)z^{mp}}{\Gamma \left(b-\frac{nr}{m}\right)\Gamma \left(\frac{nr}{m}-b+1\right)\Gamma \left(m(p+(r+1)/m)\right)}=\\ \hfill \frac{1}{\Gamma \left(b\right)}+\sum _{r=1}^m\frac{z^r}{C_r}\sum _{p=0}^{\infty }\frac{{(-1)}^{np}\Gamma \left(\frac{nr}{m}+np-b+1\right)\Gamma (r+1)z^{mp}}{\Gamma \left(\frac{nr}{m}-b+1\right)\Gamma \left(mp+r+1)\right)}\end{array}$$

We use Prop. A1 to compute

$$\begin{array}{cc}\frac{\Gamma \left(n(r/m+p)-b+1\right)}{\Gamma \left(\frac{nr}{m}-b+1\right)}=n^{np}\prod _{j=0}^{n-1}{\left(\frac{r}{m}+\frac{1-b}{n}+\frac{j}{n}\right)}_p=\hfill & \\ & \hfill n^{np}\prod _{j=n-1}^0\underset{b_j^{\prime }}{\underbrace{{\left(1+\frac{r}{m}-\frac{j+b}{n}\right)}_p}}\end{array}$$

Finally, we read off the parameters $b_j^{\prime }=1+r/m-(b+j)/n$ with an increment $1/n$. Then we can formulate the following

Theorem 2

(Second HG Representation). For $b\le 1$ and $n\le m$ non-negative co-prime integers, $a=-n/m$,

$$W\left(\left.-n/m,b\right|z\right)=\sum _{r=0}^{m-1}\frac{z^r}{r!\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1,{\overrightarrow{b}}^{\prime }\\ \overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{{(-)}^n{z}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]$$

(18)

where ${\overrightarrow{b}}^{\prime }=\{b_0^{\prime }\dots b_{n-1}^{\prime }\}$, $\overrightarrow{c}=\{c_0\dots c_{m-1}\}$ and

$$b_j^{\prime }=1+r/m-(b+j)/n,c_j=(r+1+j)/m$$

Observe that for $r=m-1$ $c_1=1$ therefore, the GHG functions reduce to ${}_{n}F_{m-1}$. For $b\ge 1$ a polynomial part P must be also added to the representation as follows.

Theorem 3

(Third HG representation). Suppose that a and b are rational parameters and $b\ge 1$ and $\left|a\right|\le 1$. Define the polynomial $P_b(-a,z)$ by the integral recursion

$$P_b(-a,z)={\int }_0^z{P}_{b-a}(-a,x)dx+c_{b-1},$$

(19)

where $c_{b-1}=1/(b-1)!$ if b is an integer and 0 otherwise. Furthermore, define $P_0(z,-a):=1$ and for $b<0$ assign $P_b(z,-a):=0$ identically. Then for $a=-n/m$ and $b\ne 0$

$$W\left(\left.-n/m,b\right|z\right)=\sum _{r=0}^{m-1}\frac{z^r}{r!\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1,{\overrightarrow{b}}^{\prime }\\ \overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{{(-)}^n{z}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]+P_b(-a,z)$$

(20)

where m and n are co-prime numbers.

Proof. 

First we prove that the arc integral results in a polynomial in z. Suppose that $b\ge 1$ is rational and $-a=n/m$ as before. Consider the arc BCD. We change variables as $\xi =ϵ{\eta }^m$. Then the integral becomes

$$I=m{\oint }_{BCD}\frac{d\eta {ϵ}^{1-b}}{{\eta }^{(b-1)m+1}}{e}^{{ϵ}^{\frac{n}{m}}{\eta }^{n}z+ϵ{\eta }^m}$$

Development of the kernel in infinite series results in

$$ker=m\frac{d\eta {ϵ}^{1-b}}{{\eta }^{(b-1)m+1}}\sum _{j=0}^{\infty }\sum _{i=0}^j\frac{{ϵ}^{\frac{im+\left(j-i\right)n}{m}}{\eta }^{im+\left(j-i\right)n}{z}^{j-i}}{i!\left(j-i\right)!}$$

The scale-invariant part of the series is given by the members $k_j$ for which

$${ϵ}^{\frac{im+\left(j-i\right)n}{m}-b+1}=1$$

This is given by the constraint

$$i=\frac{\left(b-1\right)m-jn}{m-n}$$

Therefore,

$$k_j=\frac{d\eta m{z}^{\frac{\left(j-b+1\right)n}{n-m}}}{\eta \left(\frac{\left(b-1\right)n-jm}{n-m}\right)!\left(\frac{\left(j-b+1\right)n}{n-m}\right)!}$$

Changing again variables to $\eta =e^{i\phi /m}$ results in the integral

$$\begin{array}{cc}{c}_j=\frac{1}{2\pi i}\frac{i{z}^{\frac{\left(j-b+1\right)m}{m-n}}}{\left(\frac{\left(b-1\right)m-jn}{m-n}\right)!\left(\frac{\left(j-b+1\right)m}{m-n}\right)!}·{\int }_{-\pi }^{\pi }{e}^{\frac{i\left(bn-1\right)\phi }{n}+\frac{i\phi }{n}-ib\phi }d\phi =\hfill & \\ & \hfill \frac{z^{\frac{\left(j-b+1\right)m}{m-n}}}{\left(\frac{\left(b-1\right)m-jn}{m-n}\right)!\left(\frac{\left(j-b+1\right)m}{m-n}\right)!}\end{array}$$

Furthermore, the valid indices are given by the set

$$j:\left(\frac{\left(j-b+1\right)m}{m-n}\in \mathbb{N}\right)\cup \left(\frac{\left(b-1\right)m-jn}{m-n}\in \mathbb{N}\right)$$

Equivalently, in the a-notation

$$c_j=\frac{z^{\frac{j-b+1}{1-a}}}{\left(\frac{j-b+1}{1-a}\right)!\left(\frac{-aj+b-1}{1-a}\right)!}$$

(21)

Therefore, $a<1$ must hold for $c_j$ not to vanish.

On the other hand,

$$b-1\le j\le (b-1)/a\cup j\in \mathbb{N}$$

which is a finite set. Therefore, for a rational b the integral I is a polynomial in z.

To derive the polynomial recursion we proceed as follows. By eq. 2

$$W\left(\left.-a,b\right|z\right)={\int }_0^{z}W\left(\left.-a,b-a\right|z\right)+\frac{1}{\Gamma \left(b\right)}$$

(22)

so that the equation defines a recursion relationship.

Observe that for $j=b-1\in \mathbb{N}$ the coefficient becomes

$$c_{b-1}=\frac{1}{(b-1)!}=\frac{1}{\Gamma \left(b\right)}$$

Therefore, for non-integer b there are no constant monomials. Furthermore, consider the monomial $c_j$ as a function of b. Differentiating eq. 21 one obtains the recursion

$$\begin{array}{c}\hfill \frac{d}{dz}{c}_j\left(b\right)=\frac{z^{\frac{j-b+1}{1-a}-1}}{\left(\frac{j-b+1}{1-a}-1\right)!\left(\frac{-aj+b-1}{1-a}\right)!}=\frac{z^{\frac{j-(b-a)}{1-a}}}{\left(\frac{j-(b-a)}{1-a}\right)!\left(\frac{aj-b+1}{a-1}\right)!}=c_{j-1}(b-a),\end{array}$$

which is also consistent with the integral eq. 22. Therefore, the polynomial $P_b(-a,z)$ should obey the above recursion. The second argument of the Wright function mutates and therefore it is convenient that it indexes the polynomial. □

For integer values of a, that is, when $m=1$, Th. 3 corresponds with the polynomial representation since the hypergeometric sum disappears.

6. The Special Case $b=0$

The case whenever $b=0$ needs separate treatment. From the theory of the FW functions we can formulate the following proposition.

Proposition 1.

Whenever $b=0$ we have the special FW representation

$$W\left(\left.a,0\right|z\right)=\frac{z}{\Gamma \left(a\right)}\left[\begin{array}{ccccc}\begin{array}{c}1\\ 2\end{array}& \mid & \begin{array}{c}-\\ (a,a)\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]$$

(23)

Proof. 

The proof follows by direct computation:

$$W\left(\left.a,0\right|z\right)=\sum _{k=1}^{\infty }\frac{z^k}{\Gamma \left(ak\right)\Gamma (k+1)}=\sum _{j=0}^{\infty }\frac{z^{j+1}}{\Gamma (aj+a)\Gamma (j+2)}=\frac{z}{\Gamma \left(a\right)}\left[\begin{array}{ccccc}\begin{array}{c}1\\ 2\end{array}& \mid & \begin{array}{c}-\\ (a,a)\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]$$

□

This result can be represented for rational a using the theory developed so-far as follows.

Proposition 2.

Whenever $a=m/n>0$ with $n,m$ co-prime natural numbers

$$W\left(\left.a,0\right|z\right)=\sum _{r=0}^{m-1}\frac{z^{r+1}}{(r+1)!\Gamma \left(a+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b},\overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{z^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]$$

(24)

where $\overrightarrow{b}$ has n components and $\overrightarrow{c}$ has m components given by

$$b_j=(r+1)/m+j/n,c_j=(r+2+j)/m,$$

respectively.

Proof. 

Starting from eq. 14 we observe that

$$C_r=\Gamma (ar+a)\Gamma (r+2)$$

Furthermore,

$$\frac{\Gamma \left(mp+r+2\right)}{\Gamma (r+2)}=m^{mp}\prod _{j=0}^{m-1}\underset{c_j}{\underbrace{{\left(\frac{r+2}{m}+\frac{j}{m}\right)}_p}}$$

(25)

by Prop. A1. From where we read off the component

$$c_0=\frac{r+2}{m}$$

with an increment $1/m$. □

In a similar way, we can state

Proposition 3.

Whenever $-1<a=n/m<0$ with $n,m$ co-prime natural numbers

$$W\left(\left.-n/m,0\right|z\right)=\sum _{r=0}^{m-1}\frac{z^{r+1}}{(r+1)!\Gamma \left(a+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1,{\overrightarrow{b}}^{\prime }\\ \overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{{(-)}^n{z}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]$$

(26)

where ${\overrightarrow{b}}^{\prime }=\{b_0^{\prime }\dots b_{n-1}^{\prime }\}$, $\overrightarrow{c}=\{c_0\dots c_{m-1}\}$ and

$$b_j^{\prime }=1+(r+1)/m-j/n{c}_j=(r+2+j)/m$$

Proof. 

Use the Gamma reflection formula to obtain

$$b_j^{\prime }=1-(-r/m+(-n/m+j)/n)=1+r/m-(-n/m+j)/n=1+(r+1)/m-j/n$$

□

7. Representations of the Wright Function of the First Type

The following representations can be computed using Th. 2:

7.1. Representations for $a=1/2$

The following representation holds.

$$\begin{array}{c}\hfill W\left(\left.1/2,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}\left[\begin{array}{ccc}\begin{array}{c}-\\ b,1/2\end{array}& \mid & \begin{array}{c}-\frac{z^2}{4}\hfill \\ \end{array}\end{array}\right]+\frac{z}{\Gamma (b+1/2)}\left[\begin{array}{ccc}\begin{array}{c}-\\ b+1/2,3/2\end{array}& \mid & \begin{array}{c}-\frac{z^2}{4}\hfill \\ \end{array}\end{array}\right]\end{array}$$

(27)

Here it makes sense to discuss odd and even functions.

7.2. Representations for $a=1/3$ and $a=2/3$

$$\begin{array}{cc}W\left(\left.1/3,b\right|z\right)=\frac{{}_{0}F_3\left(-;b+\frac{2}{3},\frac{4}{3},\frac{5}{3};\frac{z^3}{27}\right)z^2}{2\Gamma \left(b+\frac{2}{3}\right)}+\hfill & \\ & \hfill \frac{{}_{0}F_3\left(-,b+\frac{1}{3},\frac{2}{3},\frac{4}{3};\frac{z^3}{27}\right)z}{\Gamma \left(b+\frac{1}{3}\right)}+\frac{{}_{0}F_3\left(-;b,\frac{1}{3},\frac{2}{3};\frac{z^3}{27}\right)}{\Gamma \left(b\right)}\end{array}$$

(28)

$$\begin{array}{c}W\left(\left.2/3,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}\left[\begin{array}{ccc}\begin{array}{c}-\\ b/2,b/2+1/2,1/3,2/3\end{array}& \mid & \begin{array}{c}\frac{z^3}{108}\hfill \\ \end{array}\end{array}\right]+\hfill \\ \frac{z}{\Gamma (b+2/3)}\left[\begin{array}{ccc}\begin{array}{c}-\\ b/2+1/3,b/2+5/6,2/3,4/3\end{array}& \mid & \begin{array}{c}\frac{z^3}{108}\hfill \\ \end{array}\end{array}\right]+\\ \hfill \frac{z^2}{2\Gamma (b+4/3)}\left[\begin{array}{ccc}\begin{array}{c}-\\ b/2+2/3,b/2+7/6,4/3,5/3\end{array}& \mid & \begin{array}{c}\frac{z^3}{108}\hfill \\ \end{array}\end{array}\right]\end{array}$$

(29)

7.3. Relationship to Trigonometric and Bessel Functions

In a similar way as for the Bessel functions for half-integer values of the second parameter the Wright function can be represented by trigonometric functions as follows:

$$W\left(\left.1,1/2\right|\frac{x^2}{4}\right)=\frac{cosh\left(x\right)}{\sqrt{\pi }}$$

$$W\left(\left.1,1/2\right|-\frac{x^2}{4}\right)=\frac{cos\left(x\right)}{\sqrt{\pi }}$$

and

$$W\left(\left.1,3/2\right|\frac{x^2}{4}\right)=2\frac{sinh\left(x\right)}{\sqrt{\pi }x}$$

$$W\left(\left.1,3/2\right|-\frac{x^2}{4}\right)=2\frac{sin\left(x\right)}{\sqrt{\pi }x}$$

For $b=0$ according to eq. 23 we have the special cases

$$W\left(\left.1,0\right|x\right)=I_1\left(2\sqrt{x}\right)\sqrt{x}$$

and

$$W\left(\left.1,0\right|-x\right)=J_1\left(2\sqrt{x}\right)\sqrt{x}$$

8. Representations of the Wright Function of the Second Type

The main application of Th. 3 is the representation of the Mainardi’s function [15]

$$M_a\left(z\right)=W\left(\left.-a,1-a\right|-z\right)$$

The integral of the function is

$$I{M}_a\left(z\right)=-W\left(\left.-a,1\right|-z\right)$$

and its n^th derivative is

$$M_a^{\left(n\right)}\left(z\right)={(-)}^{n}W\left(\left.-a,1-(n+1)a\right|-z\right)$$

8.1. Representations for $a=-1/4$

$$\begin{array}{cc}{M}_{1/4}\left(z\right)=W\left(\left.-1/4,3/4\right|-z\right)=\frac{{}_{0}F_2\left(-;\frac{5}{4},\frac{3}{2};-\frac{z^4}{256}\right)z^2}{2\Gamma \left(\frac{1}{4}\right)}\hfill & \\ & \hfill -\frac{{}_{0}F_2\left(-;\frac{3}{4},\frac{5}{4};-\frac{z^4}{256}\right)z}{\sqrt{\pi }}+\frac{{}_{0}F_2\left(-;\frac{1}{2},\frac{3}{4};-\frac{z^4}{256}\right)}{\Gamma \left(\frac{3}{4}\right)}\end{array}$$

(30)

8.2. Representations for $a=-1/3$

The general formula for $b\le 1$ reads

$$\begin{array}{cc}W\left(\left.-1/3,b\right|z\right)=\frac{{}_{1}F_2\left(\frac{5}{3}-b;\frac{4}{3},\frac{5}{3};-\frac{z^3}{27}\right)z^2}{2\Gamma \left(b-\frac{2}{3}\right)}+\hfill & \\ & \hfill \frac{{}_{1}F_2\left(\frac{4}{3}-b;\frac{2}{3},\frac{4}{3};-\frac{z^3}{27}\right)z}{\Gamma \left(b-\frac{1}{3}\right)}+\frac{{}_{1}F_2\left(1-b;\frac{1}{3},\frac{2}{3},-\frac{z^3}{27}\right)}{\Gamma \left(b\right)}\end{array}$$

(31)

For $b=1/3$ and $z>0$ the equation for reduces to

$$\begin{array}{cc}W\left(\left.-1/3,1/3\right|-z\right)=-\frac{z}{3}\left(I_{2/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)-I_{-2/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)\right)=\hfill & \\ & \hfill \frac{z}{\sqrt{3}\pi }{K}_{2/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)=-\sqrt[3]{3}{Ai}^{\prime }\left(\frac{z}{\sqrt[3]{3}}\right)\end{array}$$

(32)

while

$$W\left(\left.-1/3,1/3\right|z\right)=-\frac{z}{3}\left(J_{2/3}\left(\frac{2z^{3/2}}{3^{3/2}}\right)-J_{-2/3}\left(\frac{2z^{3/2}}{3^{3/2}}\right)\right)=-\sqrt[3]{3}{Ai}^{\prime }\left(-\frac{z}{\sqrt[3]{3}}\right)$$

Regarding the Mainardi function $M_{1/3}=W\left(\left.-1/3,2/3\right|-z\right)$ eq. 31 simplifies as expected for $b=2/3$ to the Airy Aifunction, which can be represented as a weighted sum of Bessel J or I functions, respectively. That is, for $z>0$

$$\begin{array}{cc}W\left(\left.-1/3,2/3\right|-z\right)=\frac{I_{-1/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)\sqrt{z}}{\sqrt{3}}-\frac{I_{1/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)\sqrt{z}}{\sqrt{3}}=\hfill & \\ & \hfill \frac{K_{1/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)\sqrt{z}}{\pi }=\sqrt[3]{3^2}Ai\left(-z/\sqrt[3]{3}\right)\end{array}$$

(33)

while for $z<0$

$$\begin{array}{c}\hfill W\left(\left.-1/3,2/3\right|z\right)=\frac{J_{-1/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)\sqrt{z}}{\sqrt{3}}+\frac{J_{1/3}\left(\frac{2z^{\frac{3}{2}}}{3^{\frac{3}{2}}}\right)\sqrt{z}}{\sqrt{3}}=\sqrt[3]{3^2}Ai\left(-z/\sqrt[3]{3}\right)\end{array}$$

(34)

8.3. Representations for $a=-1/2$

For $b\le 1$ we have the general representation

$$W\left(\left.-1/2,b\right|z\right)=\frac{{}_{1}F_1\left(\frac{3}{2}-b;\frac{3}{2};-\frac{z^2}{4}\right)z}{\Gamma \left(b-\frac{1}{2}\right)}+\frac{{}_{1}F_1\left(1-b;\frac{1}{2};-\frac{z^2}{4}\right)}{\Gamma \left(b\right)}$$

(35)

For $b=3/4$, $z>0$

$$W\left(\left.-1/2,3/4\right|-z\right)=\frac{\sqrt{z}}{\sqrt{2}\pi }{K}_{1/4}\left(\frac{z^2}{8}\right)e^{-\frac{z^2}{8}}$$

In particular, for $b=1$ the above equation reduces to

$$W\left(\left.-1/2,1\right|z\right)=1+\mathrm{erf}\left(\frac{z}{2}\right)=erfc\left(-\frac{z}{2}\right)$$

in accordance with the polynomial reduction.

The Gaussian derivatives can be represented as

$${\left(\frac{d}{dz}\right)}^n\frac{e^{-z^2/4}}{\sqrt{\pi }}=W\left(\left.-\frac{1}{2},\frac{1-n}{2}\right|z\right)$$

(36)

Plots are presented in Figure 2. The anti-derivatives of the Gaussian kernel can be computed in a similar way using Th. 3. For example, for $b=7/2$

$$\begin{array}{cc}W\left(\left.-1/2,7/2\right|z\right)=\frac{1}{\sqrt{\pi }}\left(\frac{z^4}{60}+\frac{3z^2}{10}+\frac{8}{15}\right)e^{-\frac{z^2}{4}}\hfill & \\ & \hfill +\left(1+\mathrm{erf}\left(\frac{z}{2}\right)\right)\left(\frac{z^5}{120}+\frac{z^3}{6}+\frac{z}{2}\right)\end{array}$$

(37)

8.4. Representations for $a=-2/3$

The Mainardi function for $a=2/3$ can be represented as the difference of two exponentially-weighted Bessel K functions on the entire real line:

$$\begin{array}{c}\hfill W\left(\left.-2/3,1/3\right|-z\right)=\frac{K_{2/3}\left(-\frac{2z^3}{27}\right)z^2{e}^{-\frac{2z^3}{27}}}{3^{\frac{3}{2}}\pi }-\frac{K_{1/3}\left(-\frac{2z^3}{27}\right)z^2{e}^{-\frac{2z^3}{27}}}{3^{\frac{3}{2}}\pi }\end{array}$$

(38)

On the other hand,

$$W\left(\left.-2/3,1/3\right|z\right)=-\frac{e^{\frac{2z^3}{27}}\left(3{Ai}^{\prime }\left(\frac{z^2}{3^{\frac{4}{3}}}\right)+\sqrt[3]{3}zAi\left(\frac{z^2}{3^{\frac{4}{3}}}\right)\right)}{3^{\frac{2}{3}}}$$

in terms of the Airy function and its derivative. Plots are presented in Figure 3.

For $b=2/3$

$$W\left(\left.-2/3,2/3\right|z\right)=\frac{K_{1/3}\left(\frac{2z^3}{27}\right)z{e}^{\frac{2z^3}{27}}}{\sqrt{3}\pi }=\sqrt[3]{9}{e}^{\frac{2z^3}{27}}Ai\left(\frac{z^2}{3^{\frac{4}{3}}}\right)$$

(39)

Plots are presented in Figure 4.

For $b=4/3$

$$W\left(\left.-2/3,4/3\right|z\right)=-\frac{\left(I_{2/3}\left(\frac{2z^3}{27}\right)+I_{1/3}\left(\frac{2z^3}{27}\right)-I_{-1/3}\left(\frac{2z^3}{27}\right)-I_{-2/3}\left(\frac{2z^3}{27}\right)\right)z^2{e}^{\frac{2z^3}{27}}}{3}$$

This can be further represented as

$$W\left(\left.-2/3,4/3\right|z\right)=9^{\frac{1}{3}}z{e}^{\frac{2z^3}{27}}Ai\left(\frac{z^2}{3^{\frac{4}{3}}}\right)-3^{\frac{4}{3}}{e}^{\frac{2z^3}{27}}{Ai}^{\prime }\left(\frac{z^2}{3^{\frac{4}{3}}}\right)$$

(40)

9. Representations of the Wright Function of the Third Type

9.1. Representations for $a=-1$

This formula was recently derived in [9] and is not anticipated in the previous literature since the parameter domain is customarily restricted to $a>-1$.

$$W\left(\left.-1,b\right|z\right)=\frac{1}{\Gamma \left(b\right)}{}_{1}F_0(1-b;-;z)=\frac{{(z+1)}^{b-1}}{\Gamma \left(b\right)}$$

(41)

9.2. Representations for $a\in {\mathbb{Z}}^{-}$

For negative integers representations can be tabulated for some cases as follows. For $n=1$:

$$1,z+1$$

$n=2$:

$$\frac{1}{2},\frac{2z+1}{2},\frac{{\left(z+1\right)}^2}{2}$$

$n=3$:

$$\frac{1}{6},\frac{6z+1}{6},\frac{6z+1}{6},\frac{{\left(z+1\right)}^3}{6}$$

$n=4$

$$\frac{1}{24},\frac{24z+1}{24},\frac{24z+1}{24},\frac{12z^2+12z+1}{24},\frac{{\left(z+1\right)}^4}{24}$$

$n=5$:

$$\frac{1}{120},\frac{120z+1}{120},\frac{120z+1}{120},\frac{60z+1}{120},\frac{60z^2+20z+1}{120},\frac{{\left(z+1\right)}^5}{120}$$

$n=6$:

$$\begin{array}{cc}\frac{1}{720},\frac{720z+1}{720},\frac{720z+1}{720},\frac{360z+1}{720},\frac{360z^2+120z+1}{720},\hfill & \\ & \hfill \frac{120z^3+180z^2+30z+1}{720},\frac{{\left(z+1\right)}^6}{720}\end{array}$$

10. The Mittag-Leffler Function as a Laplace Transform of the Wright Function

The main application of the presented results so far is related to the Mittag-Leffler function $E_{a,b}\left(z\right)$. The 2 parameter Mittag-Leffler function [16,17] under the present convention will be denoted as

$$E_{a,b}\left(z\right):=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (ak+b)}=\frac{1}{\Gamma \left(b\right)}\left[\begin{array}{ccccc}\begin{array}{c}1\\ -\end{array}& \mid & \begin{array}{c}-\\ (b,a)\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right],a>0,b\ne 0$$

This immediately gives the complex integral representation according to eq. 10

$$\begin{array}{cc}{E}_{a,b}\left(z\right)=\frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\tau }}{{\tau }^b}\left[\begin{array}{ccc}\begin{array}{c}1\\ -\end{array}& \mid & \begin{array}{c}\frac{z}{{\tau }^a}\hfill \\ \end{array}\end{array}\right]d\tau =\hfill & \\ & \hfill \frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\tau }}{{\tau }^b}\frac{d\tau }{1-\frac{z}{{\tau }^a}}=\frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{{\tau }^{a-b}{e}^{\tau }}{{\tau }^a-z}d\tau \end{array}$$

(42)

For real indices $a_i$ and $b_i$, $A>0$ and $a>0$ it was proven that [14]

$$\begin{array}{cc}\left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}(A,a)\\ \dots \end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]=\hfill & \\ & \hfill \frac{1}{\Gamma \left(A\right)}{\int }_0^{\infty }{e}^{-\tau }{\tau }^{A-1}\left[\begin{array}{ccccc}\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}& \mid & \begin{array}{c}\dots \\ \dots \end{array}& \mid & \begin{array}{c}z{\tau }^a\hfill \\ \end{array}\end{array}\right]d\tau \end{array}$$

(43)

whenever the GHG function in the integral kernel converges. Then by eq. 43 for $A=1,a=1$ it follows immediately that

$$E_{a,b}\left(z\right)={\int }_0^{\infty }{e}^{-t}W\left(\left.a,b\right|zt\right)dt$$

(44)

This representation can be used to derive also a Laplace transform pair:

$$\begin{array}{cc}\frac{1}{s}{E}_{a,b}\left(\frac{1}{s}\right)=\frac{1}{s}{\int }_0^{\infty }{e}^{-\tau }W\left(\left.a,b\right|\frac{\tau }{s}\right)d\tau =\hfill & \\ & \hfill \frac{1}{s}{\int }_0^{\infty }{e}^{-st}W\left(\left.a,b\right|\frac{st}{s}\right)d\left(st\right)={\int }_0^{\infty }{e}^{-st}W\left(\left.a,b\right|t\right)dt\end{array}$$

for $s>0$, since the integration variable $\tau =st$ is positive. Therefore,

$$W\left(\left.a,b\right|t\right)÷\frac{1}{s}{E}_{a,b}\left(\frac{1}{s}\right),a>0$$

(45)

On the other hand, for the Wright function of the second type we have

$$\begin{array}{c}{\int }_0^{\infty }{e}^{-st}W\left(\left.-a,b\right|t\right)dt=\frac{1}{2\pi i}{\int }_0^{\infty }{e}^{-st}dt{\int }_{H{a}^{-}}\frac{e^{\xi +t{\xi }^a}}{{\xi }^b}d\xi =\hfill \\ \frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\xi }}{{\xi }^b}d\xi {\int }_0^{\infty }{e}^{t({\xi }^a-s)}dt=\frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\xi }}{{\xi }^b({\xi }^a-s)}d\xi =\\ \hfill \frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\xi }{\xi }^{a-(a+b)}}{{\xi }^a-s}d\xi =E_{a,a+b}\left(s\right)\end{array}$$

under the condition $\mathrm{Re}({\xi }^a-s)<0$. Therefore, the corresponding Laplace transform pair is

$$W\left(\left.-a,b\right|t\right)÷E_{a,a+b}\left(s\right),0\le a\le 1$$

(46)

This gives the relationship between the Wright and Mittag-Leffler functions.

For every rational parameter pair the ML function is reducible to a finite sum of HG functions as the following theorem [9]:

Theorem 4

(Mittag-Leffler HG Representation). Suppose that $a=n/m>0$, where n and m are co-prime, and $b\ne 0$. Then

$$E_{a,b}\left(z\right)=\sum _{r=0}^{m-1}\frac{z^r}{\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b}\end{array}& \mid & \begin{array}{c}\frac{z^m}{n^n}\hfill \\ \end{array}\end{array}\right],$$

(47)

where $\overrightarrow{b}$ has n components

$$b_j=r/m+(b+j)/n$$

Proof. 

Starting from

$$E_{n/m,b}\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (ak+b)}=\sum _{q=0}^{m-1}\sum _{p\ge q/m}^{\infty }\frac{z^{mp-q}}{\Gamma \left(a\right(mp-q)+b)}$$

since the integer n can be partitioned as $k=mp-q$, where $q=0,\dots m-1$. After some algebra we obtain

$$E_{n/m,b}\left(z\right)=\frac{1}{\Gamma \left(b\right)}+\sum _{r=1}^m{z}^r\sum _{p=0}^{\infty }\frac{z^{mp}}{\Gamma (ap+ra+b)}.$$

Observe that for $p=0$ the inner series coefficient is

$$C_r=\Gamma (ra+b)=\Gamma (nr/m+b),$$

which serves as a normalization factor. Therefore, the series transforms as

$$E_{n/m,b}\left(z\right)=\sum _{r=0}^m\frac{z^r}{C_r}\sum _{p=0}^{\infty }\frac{C_r}{\Gamma \left(n(p+r/m)+b\right)}{z}^{mp}$$

(48)

Further, use Prop. A1 to obtain

$$\frac{\Gamma \left(n(p+r/m)+b\right)}{\Gamma (nr/m+b)}=n^{np}\prod _{j=0}^{n-1}\underset{b_j}{\underbrace{{\left(r/m+b/n+j/n\right)}_p}}$$

(49)

Therefore,

$$E_{n/m,b}\left(z\right)=\sum _{r=0}^m\frac{z^r}{\Gamma (ra+b)}\sum _{p=0}^{\infty }\frac{z^{mp}}{n^{np}{\displaystyle \prod _{j=0}^{n-1}}{b}_j}$$

From where we read off

$$b_0=\frac{r}{m}+\frac{b}{n}$$

with an increment $1/n$ so that

$$E_{n/m,b}\left(z\right)=\sum _{r=0}^{m-1}\frac{z^r}{\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b}\end{array}& \mid & \begin{array}{c}\frac{z^m}{n^n}\hfill \\ \end{array}\end{array}\right]$$

□

Observe that for $r=m-1$ $c_1=1$; therefore, the GHG functions reduce to ${}_{0}F_{m-1}$. Unlike for the Wright function, whenever $b=0$

$$E_{a,0}\left(z\right)=\sum _{k=0}^{\infty }\frac{z^{k+1}}{\Gamma (ak+a)}=\frac{z}{\Gamma \left(a\right)}\left[\begin{array}{ccccc}\begin{array}{c}1\\ -\end{array}& \mid & \begin{array}{c}-\\ (a,a)\end{array}& \mid & \begin{array}{c}z\hfill \\ \end{array}\end{array}\right]=z{E}_{a,a}\left(z\right)$$

Therefore, the previous case directly applies.

$$E_{n/m,0}\left(z\right)=z\sum _{r=0}^{m-1}\frac{z^r}{\Gamma \left(a+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b}\end{array}& \mid & \begin{array}{c}\frac{z^m}{n^n}\hfill \\ \end{array}\end{array}\right],a=n/m$$

(50)

where $\overrightarrow{b}$ has n components $b_j=(r+1)/m+j/n$.

10.1. Some Integral Identities Interlinking the ML and Wright Functions

This allows one to write the following sets of integrals by application of eq. 44: For $m,n>0$ according to the First Representation Theorem

$$\begin{array}{c}{E}_{n/m,b}\left(z\right)=\sum _{r=0}^{m-1}\frac{z^r}{\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b}\end{array}& \mid & \begin{array}{c}\frac{z^m}{n^n}\hfill \\ \end{array}\end{array}\right]=\hfill \\ {\int }_0^{\infty }{e}^{-t}\sum _{r=0}^{m-1}\frac{z^r{t}^r}{r!\Gamma \left(b+ar\right)}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b},\overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{z^m{t}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]dt=\\ \hfill \sum _{r=0}^{m-1}\frac{z^r}{r!\Gamma \left(b+ar\right)}{\int }_0^{\infty }{t}^r{e}^{-t}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b},\overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{z^m{t}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]dt\end{array}$$

for $b\ne 0$. Therefore, after the substitution $y=z^m/n^n$ we have

$$\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b}\end{array}& \mid & \begin{array}{c}y\hfill \\ \end{array}\end{array}\right]=\frac{1}{\Gamma (r+1)}{\int }_0^{\infty }{t}^r{e}^{-t}\left[\begin{array}{ccc}\begin{array}{c}1\\ \overrightarrow{b},\overrightarrow{c}\end{array}& \mid & \begin{array}{c}y\frac{t^m}{m^m}\hfill \\ \end{array}\end{array}\right]dt,$$

(51)

where $\overrightarrow{b}=\{r/m+(b+j)/n\}$, $\overrightarrow{c}=\{(r+1+j)/m\}$ as discussed above. The last formula can be used to produce many integral identities between GHG functions.

10.2. Analytical Continuation of the ML Function for Negative Parameters

The integral representation allows one to continue analytically the ML for negative first parameters Then one has

$$E_{-a,b}\left(z\right):=\frac{1}{2\pi i}{\int }_{H{a}^{-}}\frac{e^{\tau }}{{\tau }^b}\frac{d\tau }{1-z{\tau }^a},a>0$$

Therefore, for rational parameters we can apply the Second and Third Representation theorems to obtain for $\left|a\right|<1$

$$E_{-n/m,b}\left(z\right)=\sum _{r=0}^{m-1}\frac{z^r}{r!\Gamma \left(b+ar\right)}{\int }_0^{\infty }{e}^{-t}{t}^r\left[\begin{array}{ccc}\begin{array}{c}1,\overrightarrow{b}\\ \overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{{(-)}^n{z}^m{t}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]dt$$

(52)

for $\left|b\right|<1$ and

$$\begin{array}{cc}{E}_{-n/m,b}\left(z\right)=\sum _{r=0}^{m-1}\frac{z^r}{r!\Gamma \left(b+ar\right)}{\int }_0^{\infty }{e}^{-t}{t}^r\left[\begin{array}{ccc}\begin{array}{c}1,\overrightarrow{b}\\ \overrightarrow{c}\end{array}& \mid & \begin{array}{c}\frac{{(-)}^n{z}^m{t}^m}{n^n{m}^m}\hfill \\ \end{array}\end{array}\right]dt+\hfill & \\ & \hfill {\int }_0^{\infty }{e}^{-t}{P}_b(n/m,zt)dt\end{array}$$

(53)

otherwise. From where we see that the integrals for non-natural parameters do not converge as they would involve kernels of ${}_{n+1}F_0$ according to eq. 51. Therefore, the analytical continuation is defined only for negative integers $a,b$ like in the case for the Wright function. In such case, (i.e. $b=m\in \mathbb{N}$)

$$E_{-n,m}\left(z\right)={\int }_0^{\infty }{e}^{-t}{P}_m(n,zt)dt=1+\sum _{k=1}^m{c}_k{z}^k$$

which is a polynomial in z. The coefficients of this polynomial can be evaluated from the formula

$$c_k=a_k{\int }_0^{\infty }{e}^{-t}{t}^{k}dt=k!a_k$$

Some examples can be presented as follows: For $n=2$

$$1,z+1$$

For $n=3$

$$\frac{1}{2},\frac{2z+1}{2},\frac{2z^2+2z+1}{2}$$

For $n=4$

$$\frac{1}{6},\frac{6z+1}{6},\frac{6z+1}{6},\frac{6z^3+6z^2+3z+1}{6}$$

For $n=5$

$$\frac{1}{24},\frac{24z+1}{24},\frac{24z+1}{24},\frac{24z^2+12z+1}{24},\frac{24z^4+24z^3+12z^2+4z+1}{24},$$

etc. These polynomials can be rightfully called Mittag-Leffler polynomials.

11. Discussion

The original goal of the present work was to to provide the ground truth for purely numerical algorithms for the evaluation of the Wright function. Such algorithms are a subject of continuous development [5,18,19,20].

The contributions of the present work can be discussed in several directions. On the first place, from a fundamental perspective, the existence of Wright function of the third type has been overlooked in the literature. This can be probably attributed to the extant focus on the Mainardi’s function, which is not defined for $a=1$. On the second place, the present work completes all cases of finite representations of the Wright function. It should be noted that The Second and Third Representation theorems could not be traced to the literature prior to the preliminary presentation in [10]. Finally, one can also envision application in definite integration to be incorporated into different CAS integration – i.e. using eq. 51; and Inverse Laplace transform routines – i.e. using eq. 45 and 46.