Generalized Erdélyi-Kober Fractional Integrals and Images of Special Functions

Abstract

The Riemann-Liuoville fractional integrals are the simplest and most popular operators of the classical fractional calculus. But their variants, the Erdélyi-Kober operators of fractional integration, have many more applications due to the freedom to choose the additional (three) parameters. We introduce and study a generalization of the Erdélyi-Kober and Riemann-Liuoville fractional integrals, where the elementary kernel function is replaced by a suitably chosen $I_{1,1}^{1,0}$-function. The I-functions introduced by Rathie in 1997 are generalized hypergeometric functions extending the Fox H-functions and the Meijer G-functions. Note that till recently this new class of special functions has not been popular because of their too complicated structure involving fractional powers of the Gamma functions and their multi-valued behavior. However, the I-functions happened to arise not only for the needs of statistical physics, but also since they included important special functions in mathematics that were not covered by the H- and G-functions. In our previous works, as Kiryakova and Paneva-Konovska, we have shown the relations of such functions, among which are the Mittag-Leffler and Le Roy type, their multi-index variants, and others related to fractional calculus, to the I-functions. Here, we propose a new theory of generalization of the Erdélyi-Kober fractional integrals, based on the use of an I-function as a kernel. This will serve next as a base to extend our generalized multi-order fractional calculus with operators involving $I_{m,m}^{m,0}$. In this paper, we also evaluate the images under these new generalized fractional integrals of special functions of very general form. Finally, in the Conclusion section, we comment on some earlier discussions on the relations between fractal geometry and fractional calculus, nowadays already without any doubts.

1. Introduction

Here, we provide brief information on the special functions involved in this study. All these are generalized hypergeometric functions known as the Fox H-functions, Meijer G-functions and Rathie I-functions, and in particular, the ${}_{p}F_q$-, Fox–Wright ${}_p\Psi _q$ and generalized Fox–Wright ${}_p\tilde{\Psi }_q$ functions, recently introduced in our works [1,2]. The definitions below are stated as in the cited original works; therefore, we try to omit some of the details.

In particular, for the definitions of the G-, H- and I-functions defined by integrals in the complex plane of Mellin–Barnes type, let us note the pioneering role of Pincherle, who, as early as 1988, introduced definitions of generalized hypergeometric functions by the later called Mellin–Barnes integrals as an efficient tool to define and deal with the higher transcendental functions. It happened that for pure mathematicians, they facilitate the representation of these functions, and for applied mathematicians, they can be successfully adopted to compute such functions. Pincherle’s priority was acknowledged in Bateman–Erdélyi Project ([3], [Vol.1, p.49]) and described in detail by Mainardi–Pagnini [4].

Definition 1.

(Ch. Fox [5], see books [6,7,8], etc.) The Fox H-function is a generalized hypergeometric function, defined by means of a contour integral of the Mellin–Barnes type

$$H_{p,q}^{m,n}\left[z\left|\begin{array}{c}{(a_i,A_i)}_1^p\\ {(b_j,B_j)}_1^q\end{array}\right.\right]={\displaystyle \frac{1}{2\pi i}}\underset{\mathcal{L}}{\int }{\mathcal{H}}_{p,q}^{m,n}\left(s\right)z^{s}ds,$$

$$with{\mathcal{H}}_{p,q}^{m,n}\left(s\right)={\displaystyle \frac{{\displaystyle \prod _{j=1}^m}\Gamma (b_j-B_{j}s){\displaystyle \prod _{i=1}^n}\Gamma (1-a_i+A_{i}s)}{{\displaystyle \prod _{j=m+1}^q}\Gamma (1-b_j+B_{j}s){\displaystyle \prod _{i=n+1}^p}\Gamma (a_i-A_{i}s)}},$$

(1)

for a complex variable $z\ne 0$ and a contour $\mathcal{L}$ in the complex domain. The orders $(m,n,p,q)$ are non-negative integers so that $0\le m\le q$, $0\le n\le p$, the parameters $A_i,B_j$ are positive, and $a_i,b_j$, $i=1,\dots ,p;j=1,\dots ,q$ are arbitrary complex such that $A_i(b_j+l)\ne B_j(a_i-l^{\prime }-1),$$l,l^{\prime }=0,1,2,\dots ;i=1,\dots ,n;j=1,\dots ,m$. Note that the integrand ${\mathcal{H}}_{p,q}^{m,n}\left(s\right)$ with $s\mapsto -s$ is the Mellin transform of the H-function (1) (when it exists).

In the above-mentioned handbooks and many other books and surveys on special functions, the reader can find more details on the definitions, properties of the Fox H-function, and conditions on the parameters and three possible types of contours. Its behavior can be characterized by means of the following values:

$$\begin{array}{c}R={\displaystyle \prod _{i=1}^p}{A}_i^{-A_i}{\displaystyle \prod _{j=1}^q}{B}_j^{B_j};\Delta ={\displaystyle \sum _{j=1}^q}{B}_j-{\displaystyle \sum _{i=1}^p}{A}_i;\gamma =\underset{s\to \infty ,s\in {\mathcal{L}}_{i\infty }}{lim}\mathrm{Re}s,\\ \mu ={\displaystyle \sum _{j=1}^q}{b}_j-{\displaystyle \sum _{i=1}^p}{a}_i+{\displaystyle \frac{p-q}{2}};a^{\ast }={\displaystyle \sum _{i=1}^n}{A}_i-{\displaystyle \sum _{i=n+1}^p}{A}_i+{\displaystyle \sum _{j=1}^m}{B}_j-{\displaystyle \sum _{j=m+1}^q}{B}_j.\end{array}$$

(2)

Depending on these, the H-function is a function analytic of z in disks $\left|z\right|<R$ or outside them, or in some sectors, or in the whole complex plane.

For example, the H-function integral converges under the following conditions: (1) $\mathcal{L}={\mathcal{L}}_{i\infty }$: $a^{\ast }>0,|argz|<a^{\ast }\pi /2$; (2) $\mathcal{L}={\mathcal{L}}_{i\infty }$: $a^{\ast }\ge 0$, $|argz|=a^{\ast }\pi /2$, $\gamma \Delta <-1-\mathrm{Re}\mu$; (3) $\mathcal{L}={\mathcal{L}}_{-i\infty }$: $\Delta >0,\left|z\right|<\infty$, or $\Delta =0,\left|z\right|<R$, or $\Delta =0,a^{\ast }\ge 0$, $\left|z\right|=R$, $\mathrm{Re}\mu <0$; (4) $\mathcal{L}={\mathcal{L}}_{+i\infty }$: $\Delta <0,\left|z\right|<\infty$, or $\Delta =0,\left|z\right|>R$, or $\Delta =0,a^{\ast }\ge 0$, $\left|z\right|=R$, $\mathrm{Re}\mu <0$. The contour ${\mathcal{L}}_{-i\infty }$ (resp. ${\mathcal{L}}_{+i\infty }$) is a left (resp. right) loop in some horizontal strip beginning at the point $-\infty +i{\phi }_1$ (resp. $+\infty +i{\phi }_1$), keeping the poles of the functions $\Gamma (b_j+B_{j}s),$ $j=1,2,\dots ,m$ on the left side, and those of $\Gamma (1-a_i-A_{i}s),i=1,2,\dots ,n$ on its right side, and ending at $-\infty +i{\phi }_2$ (resp. $+\infty +i{\phi }_2$), ${\phi }_1<{\phi }_2$. The contour ${\mathcal{L}}_{i\infty }$ starting at $\gamma -i\infty$ and ending at $\gamma +i\infty$ is such that it separates the mentioned poles; a similar situation exists for ${\mathcal{L}}_{\pm i\infty }$.

The behavior of the H-function around the singular points $z=0$ and $z=\infty$ is described in the mentioned literature, and especially for the more complicated case of the third singular point $z=R$, one can see the recent survey by Karp [9], based on the older results of Braaksma, as well as in Kiryakova ([10], [Ch.5]) when $R=1$.

The H-function reduces to the simpler Meijer G-function (C.S. Meijer [11]); see details in ([3], [Vol.1]) and in many other handbooks. If $\forall A_i=B_j=1,i=1,\dots ,p;j=1,\dots ,q$, namely,

$$\begin{gathered} H_{p,q}^{m,n}\left[z\left|\begin{array}{c}{(a_i,1)}_1^p\\ {(b_j,1)}_1^q\end{array}\right.\right]=G_{p,q}^{m,n}\left[z\left|\begin{array}{c}{\left(a_i\right)}_1^p\\ {\left(b_j\right)}_1^q\end{array}\right.\right]={\displaystyle \frac{1}{2\pi i}}\underset{\mathcal{L}}{\int }{\mathcal{G}}_{p,q}^{m,n}\left(s\right)z^{s}ds \\ ={\displaystyle \frac{1}{2\pi i}}\underset{\mathcal{L}}{\int }{\displaystyle \frac{{\displaystyle \prod _{j=1}^m}\Gamma (b_j-s){\displaystyle \prod _{i=1}^n}\Gamma (1-a_i+s)}{{\displaystyle \prod _{j=m+1}^q}\Gamma (1-b_j+s){\displaystyle \prod _{i=n+1}^p}\Gamma (a_i-s)}}{z}^{s}ds,z\ne 0. \end{gathered}$$

(3)

The behavior of the G-function can be described by the parameters in (2), taking $R=1$, $\Delta =q-p$, $\delta =m+n-{\displaystyle \frac{p+q}{2}}$. The G-function is much simpler than the H-function but yet it is enough general to include most of the Classical Special Functions (also called Named SF, SF of Mathematical Physics), the orthogonal polynomials, and many elementary functions. Such numerous particular cases are envisaged, for example, in ([3], [Vol.1]), ([10], [App. C]), [12], etc.

Among the Special Functions of Fractional Calculus (SF of FC) that are Fox H-functions but cannot be reduced to Meijer G-functions in the general case, and one of the most popular and frequently used in fractional order models are the Wright generalized hypergeometric functions ${}_p\Psi _q$ and the simplest but very important Mittag-Leffler function, known as the Queen function of the fractional calculus (refer to [13]). More details can be seen, for example, in [1,2,12], etc. We recall their definitions below.

Definition 2.

(see, e.g., [6,7], ([10], [App.E]), [13], etc.) The Wright (Fox–Wright) generalized hypergeometric function ${}_p\Psi _q\left(z\right)$ is defined by the power series:

$${}_p\Psi _q\left[\left.\begin{array}{c}(a_1,A_1),\dots ,(a_p,A_p)\\ (b_1,B_1),\dots ,(b_q,B_q)\end{array}\right|z\right]=\sum _{k=0}^{\infty }{\displaystyle \frac{\Gamma (a_1+k{A}_1)\dots \Gamma (a_p+k{A}_p)}{\Gamma (b_1+k{B}_1)\dots \Gamma (b_q+k{B}_q)}}{\displaystyle \frac{z^k}{k!}},$$

(4)

and it can be also represented as an H-function:

$$=H_{p,q+1}^{1,p}\left[-z\left|\begin{array}{c}(1-a_1,A_1),\dots ,(1-a_p,A_p)\\ (0,1),(1-b_1,B_1),\dots ,(1-b_q,B_q)\end{array}\right.\right].$$

(5)

Depending on the values in conditions (2), the function (4)–(5) is an entire function of z if $\Delta >-1$ or its series is absolutely convergent in the disk $\left\{\right|z|<R\}$ for $\Delta =-1$, the same for $\left|z\right|=R$ but if $Re\left(\mu \right)>1/2$.

For all $A_1=\dots =A_p=1,B_1=\dots =B_q=1$, the Fox–Wright function is the more popular ${}_{p}F_q$-function, representable as a Meijer G-function (3); see the old handbook ([3], [Vol.1]):

$${}_p\Psi _q\left[\left.\begin{array}{c}(a_1,1),\dots ,(a_p,1)\\ (b_1,1),\dots ,(b_q,1)\end{array}\right|z\right]=c{}_{p}F_q(a_1,\dots ,a_p;b_1,\dots ,b_q;z)$$

$$=c\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a_1\right)}_k\dots {\left(a_p\right)}_k}{{\left(b_1\right)}_k\dots {\left(b_q\right)}_k}}{\displaystyle \frac{z^k}{k!}}=G_{p,q+1}^{1,p}\left[-z\left|\begin{array}{c}1-a_1,\dots ,1-a_p\\ 0,1-b_1,\dots ,1-b_q\end{array}\right.\right];$$

(6)

$$\mathrm{with}c=\left[\prod _{i=1}^p\Gamma \left(a_i\right)/\prod _{j=1}^q\Gamma \left(b_j\right)\right],{\left(a\right)}_k:=\Gamma (a+k)/\Gamma \left(a\right).$$

Very important cases of (4), with respect to fractional calculus, are the already mentioned Mittag-Leffler (M-L) functions

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+1)}},E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}},\alpha >0,\beta >0,$$

(7)

which are the simplest entire functions of order $\rho =1/\alpha$ and type 1. The three-parameter variant is usually called the Prabhakar function (see [13] and ([1], (8))):

$$E_{\alpha ,\beta }^{\tau }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\tau \right)}_k}{\Gamma (\alpha k+\beta )}}{\displaystyle \frac{z^k}{k!}},\alpha ,\beta ,\tau \in \mathbb{C},\mathrm{Re}\alpha >0;$$

(8)

with the Pochhammer symbol ${\left(\tau \right)}_0=1,{\left(\tau \right)}_k=\Gamma (\tau +k)/\Gamma \left(\tau \right)$. For $\tau =1$, it is the M-L function $E_{\alpha ,\beta }$, and if additionally $\beta =1$, we have $E_{\alpha }$.

The multi-index Mittag-Leffler and Mittag-Leffler–Prabhakar functions by Kiryakova and Paneva–Konovska (see [1,12], etc.) are shown to be important extensions of (7) and (8), and cases of the Fox–Wright generalized hypergeometric function ${}_p\Psi _q\left(z\right)$. Let us also mention the multi-variable and multinomial variants of the Mittag-Leffler-type functions, appearing as solutions of multi-term fractional differential equations (for example, [14,15], etc.)

However, there are many other special functions playing an important role in analysis, number theory, statistics, physics, and various branches of applied mathematics that still have not been incorporated in the class of Special Functions of Fractional Calculus (SF of FC, in the sense of this notion from Kiryakova [12]). Such examples are the Riemann Zeta function, Hurwitz Zeta function, Le Roy-type functions, Feynman integrals, and their multi-index extensions. In our recent survey [1], we have brought attention to the so-called I-functions of Rathie [16], their origin from applications in statistical physics, and Feynman integrals, as well as their simpler variants such as the $\overline{H}$-functions of Inayat–Hussain [17] (see also [18,19,20]). These are further extensions of the classes of the Wright generalized hypergeometric functions ${}_p\Psi _q$ (4) and of the Fox H-functions (1), but they have somehow been neglected till recently, considered too complicated or artificial innovations. Our aim was to provide new insights into the I-functions, confirm their role in enriching the class of the SF of FC, and explore them as kernels of new operators of Generalized Fractional Calculus.

Definition 3.

The I-function was defined by Rathie [16] by a contour integral in the complex domain of the kind of Mellin–Barnes:

$$I_{p,q}^{m,n}\left[z\left|\begin{array}{c}{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q\end{array}\right.\right]={\displaystyle \frac{1}{2\pi i}}\underset{\mathcal{L}}{\int }{\mathcal{I}}_{p,q}^{m,n}\left(s\right)z^{s}ds,z\ne 0,$$

(9)

where the integrand function

$${\mathcal{I}}_{p,q}^{m,n}\left(s\right)={\displaystyle \frac{{\displaystyle \prod _{k=1}^m}{\Gamma }^{{\beta }_k}(b_k-B_{k}s){\displaystyle \prod _{j=1}^n}{\Gamma }^{{\alpha }_j}(1-a_j+A_{j}s)}{{\displaystyle \prod _{k=m+1}^q}{\Gamma }^{{\beta }_k}(1-b_k+B_{k}s){\displaystyle \prod _{j=n+1}^p}{\Gamma }^{{\alpha }_j}(a_j-A_{j}s)}}$$

(10)

contains Gamma functions with some power exponents, ${\alpha }_j$, $j=1,\dots ,p$, and ${\beta }_k$, $k=1,\dots ,q$, that, in general, can be arbitrary positive numbers, not obligatory integer ones. The possible contours $\mathcal{L}$ are discussed briefly below, following the original work by Rathie [16].

Note that for non-integer values of the “power”-parameters ${\alpha }_j$ and ${\beta }_k$, the I-function (9) cannot be reduced to an H-function or to other “classical” special functions. But if some groups of ${\alpha }_j$’s and ${\beta }_k$’s are taken to be 1, it is the simpler $\overline{H}$-function of Inayat–Hussain [17] (also see [18,19,20] for more details on the possible contours, some results and applications).

In the above definition, there are non-negative integer orders $0\le m\le q$, $0\le n\le p$, parameters $A_i>0$ and $B_j>0$ and $a_j,j=1,\dots ,p$, $b_k,k=1,\dots ,q$ that can be complex numbers such that no singularity of ${\Gamma }^{{\beta }_k}(b_k-B_{k}s),k=1,\dots ,m$ coincides with any singularity of ${\Gamma }^{{\alpha }_j}(1-a_j+A_{j}s),j=1,\dots ,n$.

In general, the I-function is a multi-valued function of the complex variable z and has a branch point at $z=0$. We suppose $z^s=exp\left[s\right\{\log \left|z\right|+i\arg z\left\}\right]$. In general, for non-integers ${\alpha }_j$ and ${\beta }_k$, the mentioned singularities are no longer poles but are just branching points of the multi-valued fractional powers of the Gamma functions. The branch cuts in the complex plane are made in a way so that the integration’s path can be distorted to one of the contours, as described in detail in the preliminary paper of Rathie [16]:

  (a) $\mathcal{L}$ goes from $c-i\infty$ to $c+i\infty$, with suitably chosen real c allowing the singularities of ${\Gamma }^{{\beta }_k}(b_k-B_{k}s),k=1,\dots ,m,$ to lie to the right of $\mathcal{L}$, and all singularities of ${\Gamma }^{{\alpha }_j}(1-a_j+A_{j}s),j=1,\dots ,n,$ to lie to the left of $\mathcal{L}$;

  (b) $\mathcal{L}$ is a loop that begins and ends at $+\infty$ and encircles, once in the clockwise direction, all the singularities of ${\Gamma }^{{\beta }_k}(b_k-B_{k}s),k=1,\dots ,m$, but none of the singularities of ${\Gamma }^{{\alpha }_j}(1-a_j+A_{j}s),j=1,\dots ,n$;

  (c) $\mathcal{L}$ is a loop beginning and ending at $-\infty$ and encircling, once in the anticlockwise direction, all the singularities of ${\Gamma }^{{\alpha }_j}(1-a_j+A_{j}s),j=1,\dots ,n$, but none of the singularities of ${\Gamma }^{{\beta }_k}(b_k-B_{k}s),k=1,\dots ,m$.

According to Rathie [16], the contour (a) can be interpreted as a case of the contours (c) and (b), while compared to (c), (b) is just with a changed sign of the function and direction. Hence, the above-defined contours may lead to the same result, in the case when they are possible.

The conditions that describe the behavior and the properties of the I-function are given by the following parameters (and are rather analogous to these for the H-function in (2)):

$$\begin{array}{c}\Delta ={\displaystyle \sum _{k=1}^m}{\beta }_k{B}_k-{\displaystyle \sum _{k=m+1}^q}{\beta }_k{B}_k+{\displaystyle \sum _{j=1}^n}{\alpha }_j{A}_j-{\displaystyle \sum _{j=n+1}^p}{\alpha }_j{A}_j;\\ \mu ={\displaystyle \sum _{k=1}^q}{\beta }_k{B}_k-{\displaystyle \sum _{j=1}^p}{\alpha }_j{A}_j;\\ \nabla ={\displaystyle \sum _{j=1}^p}{\alpha }_j[Re\left(a_j\right)-1/2]-{\displaystyle \sum _{k=1}^q}{\beta }_k[Re\left(b_k\right)-1/2];\\ R={\displaystyle \prod _{k=1}^q}{B}_k^{{\beta }_k{B}_k}/{\displaystyle \prod _{j=1}^p}{A}_j^{{\alpha }_j{A}_j}.\end{array}$$

(11)

According to Rathie [16], the I-function (9) with contour $\mathcal{L}$ of the type (a), converges when $|argz|<\Delta \pi /2$, if $\Delta >0$, while for $|argz|=\Delta \pi /2$, $\Delta \ge 0$, this integral converges absolutely in the following cases: (i) $\mu =0$ and $\nabla >1$; (ii) $\mu \ne 0$, if $s=\sigma +it$, with real $\sigma$ and t, and $\sigma$ is so that $\nabla +\sigma \mu >1$ for $\left|t\right|\to \infty$; (iii) the integral (9) with contour $\mathcal{L}$ of type (b) converges if $q\ge 1$ and either $\mu >0$, or $\mu =0$ inside the circle $\left|z\right|<R$.

Note that in this work, we consider (as kernels of the introduced generalized fractional integrals) only $I_{1,1}^{1,0}$-functions that fall, especially in case (iii), under

$$q\ge 1(\mathrm{here}q=1),\mu =0,\left|z\right|<R=1,\mathrm{contour}\mathcal{L}\mathrm{of} \mathrm{type} {\displaystyle \mathsf{(}}\mathbf{b}{\displaystyle \mathsf{)}}\mathrm{that} \mathrm{can} \mathrm{be} \mathrm{converted} \mathrm{into} {\displaystyle \mathsf{(}}\mathbf{a}{\displaystyle \mathsf{)}}.$$

(12)

Similarly, in the general case, the integral (9) with contour of type (c) converges if $p\ge 1$ and either $\mu <0$, or $\mu =0$ if $\left|z\right|>R$.

In ([21], [Sect.3]), Rogosin and Dubatovskaya proposed an example of how single-valued branches of the multi-valued $\Gamma$-functions with fractional powers can be fixed and suitable branch cuts drawn, in the case of a similar function denoted by $HG$ (eq.(3.1), [21]).

There, a contour $\mathcal{L}$ is taken to be of the Slater type (see Marichev ([22], [Ch.4])), following the ideas of the Slater theorem expanded for $\Gamma$-functions with arbitrary powers.

As we noted in [1], many important special functions related to fractional-order parameters and models cannot be incorporated as cases of the H-, G- and Fox–Wright generalized hypergeometric function (4)–(5) but are shown now to be cases of the I-functions. And in particular, these can be treated as cases of the newly introduced generalized hypergeometric function extending the ${}_p\Psi _q$ one.

Definition 4.

The generalized Fox–Wright function is introduced in [1,2]) by the following power series:

$${}_p\tilde{\Psi }_q\left[\left.\begin{array}{c}{\displaystyle {(a_j,A_j;{\alpha }_j)}_{j=1}^p}\\ {\displaystyle {(b_i,B_i;{\beta }_i)}_{i=1}^q}\end{array}\right|z\right]:=\sum _{k=0}^{\infty }{\displaystyle \frac{{\displaystyle \prod _{j=1}^p}{\Gamma }^{{\alpha }_j}(A_{j}k+a_j)}{{\displaystyle \prod _{i=1}^q}{\Gamma }^{{\beta }_i}(B_{i}k+b_i)}}·{\displaystyle \frac{z^k}{k!}},$$

(13)

with arbitrary ${\alpha }_j>0$, ${\beta }_i>0$, $j=1,\dots ,p$, $i=1,\dots ,q$, and characteristic parameters like in (11):

$$\tilde{\mu }=1+\sum _{i=1}^q{\beta }_i{B}_i-\sum _{j=1}^p{\alpha }_j{A}_j,\tilde{R}=\prod _{i=1}^q{\left(B_i\right)}^{{\beta }_i{B}_i}/\prod _{j=1}^p{\left(A_j\right)}^{{\alpha }_j{A}_j}.$$

(14)

Let us mention that the Fox–Wright function (4) is obtained from (13) by replacing all ${\alpha }_j=1$ and all ${\beta }_i=1$, namely,

$${}_p\Psi _q\left[\left.\begin{array}{c}(a_1,A_1),\dots ,(a_p,A_p)\\ (b_1,B_1),\dots ,(b_q,B_q)\end{array}\right|z\right]={}_p\tilde{\Psi }_q\left[\left.\begin{array}{c}(a_1,A_1,1),\dots ,(a_p,A_p,1)\\ (b_1,B_1,1),\dots ,(b_q,B_q,1)\end{array}\right|z\right].$$

(15)

As we proved in [1,2], ${}_p\tilde{\Psi }_q$ is an entire function if $\tilde{\mu }>0$, or an analytical one in $\left|z\right|<\tilde{R}$ if $\tilde{\mu }=0$. To determine single-valued branches of $z^k$ and of the multi-valued $\Gamma$-functions with arbitrary (fractional) powers, we need to insert suitable cuts in the complex plane, like it is discussed in the general case of the I-function.

Further, let us emphasize that the generalized Fox–Wright function ${}_p\tilde{\Psi }_q$ can be presented as the following I-function; see ([2], [Formula (7)]):

$${}_p\tilde{\Psi }_q\left[\left.\begin{array}{c}{(a_j,A_j,{\alpha }_j)}_{j=1}^p\\ {(b_i,B_i,{\beta }_i)}_{i=1}^q\end{array}\right|z\right]=I_{p,q+1}^{1,p}\left[-z\left|\begin{array}{c}{(1-a_j,A_j,{\alpha }_j)}_1^p\\ (0,1,1),{(1-b_i,B_i,{\beta }_i)}_1^q\end{array}\right.\right].$$

(16)

As a conclusion of this section, we note that the basic properties of the generalized Fox–Wright function ${}_p\tilde{\Psi }_q$ are given in our paper ([2], [Theorems 1 and 2]).

2. “Classical” Erdélyi-Kober Operators of FC

The most popular and basic definition in fractional calculus (FC) is for the Riemann-Liouville (R-L) integral of fractional order $\alpha >0$ ([7,23], etc.):

$$I^{\alpha }f\left(z\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^z{(z-\tau )}^{\alpha -1}f\left(\tau \right)d\tau ={\displaystyle \frac{z^{\alpha }}{\Gamma \left(\alpha \right)}}{\int }_0^1{(1-t)}^{\alpha -1}f\left(zt\right)dt.$$

(17)

For $\alpha =0$, it is assumed to be the identity operator.

Here, for brevity, we discuss only the so-called left-hand-sided variants of the FC operators, for real values of all parameters and skip the definitions of the corresponding Riemann-Liuoville and Caputo fractional derivatives and the details on the conditions in various functional spaces. For these details, one can refer to [10,24], as well as [23,25], etc.

The Erdélyi-Kober (E-K) fractional integrals of order $\alpha >0$, with a “weight” parameter $\nu \in \mathbb{R}$ and an additional parameter ${\beta }^{\ast }>0$ is among the “classical” modifications of (17) with a very wide range of applications, due to the greater freedom of choice offered by the two additional parameters.

Attention: In our previous works, we have used a denotation $\beta$, but here, we substitute it with ${\beta }^{\ast }$, just because in the generalization with an I-function kernel, we now use $\beta$ as simpler than the fractions $1/{\beta }^{\ast }$ appearing in the Generalized Fractional Calculus (GFC) operators until now, as indicated below. Then, one can further assume that $\beta =1/{\beta }^{\ast }$:

$$\begin{gathered} I_{{\beta }^{\ast }}^{\nu ,\alpha }f\left(z\right)=z^{-{\beta }^{\ast }(\nu +\alpha )}{\int }_0^z{\displaystyle \frac{{(z^{{\beta }^{\ast }}-{\tau }^{{\beta }^{\ast }})}^{\alpha -1}}{\Gamma \left(\alpha \right)}}{\tau }^{{\beta }^{\ast }\nu }f\left(\tau \right)d\left({\tau }^{{\beta }^{\ast }}\right) \\ ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^1{(1-t)}^{\alpha -1}{t}^{\nu }f\left(z{t}^{1/{\beta }^{\ast }}\right)dt \\ ={\displaystyle \frac{{\beta }^{\ast }}{\Gamma \left(\alpha \right)}}{\int }_0^1{(1-{\sigma }^{{\beta }^{\ast }})}^{\alpha -1}{\sigma }^{{\beta }^{\ast }\nu +{\beta }^{\ast }-1}f\left(z\sigma \right)d\sigma . \end{gathered}$$

(18)

For $\alpha =0$, again by default, (18) is the identity: $I_{{\beta }^{\ast }}^{\nu ,0}f\left(z\right)=f\left(z\right)$.

Note that the semigroup property, as one of the main features of FC operators, is satisfied ([10], [Ch.2]):

$$I_{{\beta }^{\ast }}^{\nu ,{\alpha }_1}{I}_{{\beta }^{\ast }}^{\nu +{\alpha }_1,{\alpha }_2}f\left(z\right)=I_{{\beta }^{\ast }}^{\nu +{\alpha }_1,{\alpha }_2}{I}_{{\beta }^{\ast }}^{\nu ,{\alpha }_1}f\left(z\right)=I_{{\beta }^{\ast }}^{\nu ,{\alpha }_1+{\alpha }_2}f\left(z\right),{\alpha }_1>0,{\alpha }_2>0.$$

(19)

The operators (18) appeared initially in works by Erdélyi, Kober, and Sneddon, with ${\beta }^{\ast }=1$, ${\beta }^{\ast }=2$, have been widely used and studied with arbitrary ${\beta }^{\ast }>0$: by Sneddon [26], later by Kiryakova [10] and subsequently, by Luchko et al. [25].

The Erdélyi-Kober fractional integral (18) has the property to preserve the power functions up to a multiplier:

$$I_{{\beta }^{\ast }}^{\nu ,\alpha }\left\{z^p\right\}={\vartheta }_p{z}^p,\mathrm{with}{\vartheta }_p={\displaystyle \frac{\Gamma (p/{\beta }^{\ast }+\nu +1)}{\Gamma (p/{\beta }^{\ast }+\nu +\alpha +1)}},p>-{\beta }^{\ast }(\nu +1),$$

(20)

where $\alpha \ge 0$, $\nu >-{\displaystyle \frac{p}{{\beta }^{\ast }}}-1$ are the conditions for the operator’s parameters.

Often, the E-K operators are considered and used in operational/convolutional calculus in the following form:

$$\mathbb{I}f\left(z\right):=z^{{\alpha }_0}{I}_{{\beta }^{\ast }}^{\nu ,\alpha }f\left(z\right),\mathrm{with}{\alpha }_0\ge 0,$$

(21)

as more typical for the operators of integration having the feature to increase the powers.

Evidently, for the particular choice $\nu =0,{\beta }^{\ast }=1$, ${\alpha }_0=\alpha$, we have the R-L integral

$$I^{\alpha }f\left(z\right)=z^{\alpha }{I}_1^{0,\alpha }f\left(z\right).$$

(22)

The theory of operators (18), (21) and the corresponding Erdélyi-Kober fractional derivatives $D_{{\beta }^{\ast }}^{\nu ,\alpha }$ and their commutative compositions lie in the base of the Generalized Fractional Calculus (GFC), as by Kiryakova [10] (also see the survey in [24]). The Erdélyi-Kober operators and their compositions have also been studied and widely used by Luchko et al.; see, e.g., [25,27]. In the works of Kiryakova [10] and ([28], [Lemmas 1-2, Th.2]), results are provided for the Erdélyi-Kober and multiple Erdélyi-Kober images of the so-called Special Functions of Fractional Calculus (SF of FC) in general cases, namely the Fox H-functions, Meijer G-functions, Fox–Wright generalized hypergeometric functions ${}_p\Psi _q$, and of all their particular cases mentioned in [12]. For Erdélyi-Kober operators and images of the Le Roy-type functions, also including Miitag–Leffler and Prabhakar types, see Kiryakova and Paneva–Konovska [1]. In this way, we proposed a unified approach (see [28]) to foresee and give explicitly the images of very wide classes of SF under the GFC operators in the sense of [10], i.e., under multiple compositions of Erdélyi-Kober operators. As very particular cases, we illustrate the results for some popular variants of FC operators, such as Saigo operators, Marichev-Saigo-Maeda, etc.

We skip the details on the various functional spaces (of weighted continuous functions, weighted Lebesgue integrable functions, or weighted analytical functions) where the theory of the Erdélyi-Kober operators has been developed ([10], [Ch.1, Ch.2, Ch.5]). The corresponding Erdélyi-Kober fractional derivatives $D_{{\beta }^{\ast }}^{\nu ,\alpha }$ such that $D_{{\beta }^{\ast }}^{\nu ,\alpha }{I}_{{\beta }^{\ast }}^{\nu ,\alpha }f\left(z\right)=f\left(z\right)$ in these spaces are introduced and studied by Kiryakova ([10], [Ch.2]) by means of explicit integro-differential expression (for Riemann–Liuoville type):

$$D_{{\beta }^{\ast }}^{\nu ,\alpha }f\left(z\right)=[\prod _{j=1}^n({\displaystyle \frac{1}{{\beta }^{\ast }}}z{\displaystyle \frac{d}{dz}}+\nu +j)]I_{{\beta }^{\ast }}^{\nu +\alpha ,n-\alpha }f\left(z\right),n-1<\alpha \le n,$$

(23)

or analogous differential-integral (for Caputo type) representation; one can see, e.g., [24,27,29].

Let us note that the kernel of the Erdélyi-Kober integral (18), the following elementary function, is an important particular case of the Meijer G-function and of the Fox H-function, but also of the Rathie I-function, namely,

$${\beta }^{\ast }{\displaystyle \frac{{(1-{\sigma }^{{\beta }^{\ast }})}^{\alpha -1}}{\Gamma \left(\alpha \right)}}{\sigma }^{{\beta }^{\ast }\nu +{\beta }^{\ast }-1}={\beta }^{\ast }{\sigma }^{{\beta }^{\ast }-1}{G}_{1,1}^{1,0}\left[{\sigma }^{{\beta }^{\ast }}\left|\begin{array}{c}\nu +\alpha \\ \nu \end{array}\right.\right]$$

$$=H_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-{\displaystyle \frac{1}{{\beta }^{\ast }}}+\alpha ,{\displaystyle \frac{1}{{\beta }^{\ast }}}\\ \nu +1-{\displaystyle \frac{1}{{\beta }^{\ast }}},{\displaystyle \frac{1}{{\beta }^{\ast }}}\end{array}\right.\right]=H_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta \\ \nu +1-\beta ,\beta \end{array}\right.\right]$$

(24)

$$=I_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,1\\ \nu +1-\beta ,\beta ,1\end{array}\right.\right].$$

In the above, we denote $\beta :=1/{\beta }^{\ast }>0$ to be in sync with the designation of the generalized E-K operator, introduced next in this paper.

Then, the E-K fractional integral $I_{{\beta }^{\ast }}^{\nu ,\alpha }$ appears as the simplest case with $m=1$ of the generalized fractional calculus operators from Kiryakova [10] with kernels $G_{m,m}^{m,0}$, resp. $H_{m,m}^{m,0}$.

Following an analogy with (24), we have introduced generalizations of the Erdélyi-Kober fractional integrals (18), taking as a kernel an $I_{1,1}^{1,0}$-function of Rathie (with an additional parameter $\gamma >0$), instead of the $G_{1,1}^{1,0}$- and $H_{1,1}^{1,0}$-functions.

3. Generalized Erdelyi-Kober Fractional Integrals with I-Function

Definition 5

(Kiryakova, Paneva-Konovska [30]). Let us take real parameters $\alpha \ge 0$, $\beta >0$, $\gamma >0$, $\nu \in \mathbb{R}$. By means of these 4 parameters and an $I_{1,1}^{1,0}$-function as a singular kernel, we define the generalized Erdélyi-Kober (gen. E-K) integral operator as

$${\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }f\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(z\sigma \right)d\sigma$$

(25)

of (fractional) order $\alpha >0$; and for $\alpha =0$ we suppose: ${\mathbb{I}}_{\beta ,\gamma }^{\nu ,0}f\left(z\right):=f\left(z\right)$.

Note the important role that the additional parameter $\gamma$ plays for the extension of the classical Erdélyi-Kober fractional integral (18) to its generalization (25). It appears as the third parameter of the involved $I_{1,1}^{1,0}$-function, a fractional power of the Gamma functions there. Compare with the representation (24). Such a parameter $\gamma >0$ is typical also for the Le Roy and Mittag-Leffler–Le Roy-type functions (37) that are some of the simplest I-functions but also appear as eigenfunctions of cases of the generalized fractional integrals from Definition 5; see details in Section 4. Specifically, for $\gamma :=1$, we have the three-parameter ($\alpha \ge 0,\beta >0$, real $\nu$) operator of the form

$$\begin{gathered} {\mathbb{I}}_{\beta ,1}^{\nu ,\alpha }f\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,1\\ \nu +1-\beta ,\beta ,1\end{array}\right.\right]f\left(z\sigma \right)d\sigma \\ =\underset{0}{\overset{1}{\int }}{H}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta \\ \nu +1-\beta ,\beta \end{array}\right.\right]f\left(z\sigma \right)d\sigma \\ ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{(1-{\sigma }^{1/\beta })}^{\alpha -1}{\sigma }^{(\nu +1)/\beta -1}f\left(z\sigma \right)d\sigma =I_{1/\beta }^{\nu ,\alpha }f\left(z\right), \end{gathered}$$

(26)

that is, the “classical” E-K fractional integral, according to definition (18) and relation (24).

First, let us analyze the above definition with respect to its kernel’s behavior, and also for the case $\alpha =0$.

For this I-function

$$I_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]={\displaystyle \frac{1}{2\pi i}}\underset{\mathcal{L}}{\int }{\displaystyle \frac{{\Gamma }^{\gamma }(\nu +1-\beta -\beta s)}{{\Gamma }^{\gamma }(\nu +1-\beta +\alpha -\beta s)}}{z}^{s}ds,$$

as mentioned in (12), the values of the parameters in (11) are as follows:

$$\Delta =\gamma \beta -\gamma \beta =0;\mu =\gamma \beta -\gamma \beta =0;\nabla =\gamma \alpha ;R={\beta }^{\gamma \beta }/{\beta }^{\gamma \beta }=1.$$

Therefore, case (iii) is applicable for the definition of this $I_{1,1}^{1,0}$ with contour $\mathcal{L}$ of the form (b), and the integral is convergent in $\left|z\right|<1$, vanishes outside the unit disc, and for $|argz|=0$, has a well-described behavior at the singular points $z=0$ and $z=1$ that ensures the convergence of (25) for functions of the form

$$f\left(z\right)=z^{\kappa }\tilde{f}\left(z\right),\tilde{f}\mathrm{continuous} (\mathrm{integrable},\mathrm{analytic})\mathrm{on}(0,\infty ),\mathrm{with}\kappa >-{\displaystyle \frac{\nu +1}{\beta }}.$$

Note that the singular points of ${\Gamma }^{\gamma }(\nu +1-\beta -\beta s)$ in the numerator are $s_k={\displaystyle \frac{\nu +1+k}{\beta }}-1>-1$, $k=0,1,2,\dots ,$ and the singularities of ${\Gamma }^{\gamma }(\nu +1-\beta +\alpha -\beta s)$ in denominator are $s_l={\displaystyle \frac{\nu +1+\alpha +l}{\beta }}-1>-1$, $l=0,1,2,\dots ,$ because of the above-assumed condition $\kappa >-{\displaystyle \frac{\nu +1}{\beta }}$ for the mentioned functional spaces. Therefore, no singularities of the involved two Gamma functions (according to general definition (10)) appear for $s<-1$. Then, a cut to ensure single values can be taken, for example, along the real half line $s>0$, and a vertical contour (type (b) can be converted into type (a)): $\mathcal{L}=(c-i\infty ,c+i\infty )$ with some $c<-1$. In this way, all the singularities are to the right of $\mathcal{L}$ that does not intersect the branch cut.

For the asymptotic behavior of the kernel $I_{1,1}^{1,0}\left(\sigma \right)$, according to (6.9) in Rathie [16], for $\sigma \to 0$, we have

$$I\left(\sigma \right)=I_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]\sim {\sigma }^c,c={\displaystyle \frac{\nu +1}{\beta }}-1,\mathrm{and} \mathrm{so},$$

$$I\left(\sigma \right){\sigma }^{\kappa }\tilde{f}\left(z\sigma \right)\sim {\sigma }^d,d={\displaystyle \frac{\nu +1}{\beta }}-1+\kappa >{\displaystyle \frac{\nu +1}{\beta }}-1-{\displaystyle \frac{\nu +1}{\beta }}=-1,$$

since $\tilde{f}$ is bounded. Near the other singularity $\sigma =1$, the behavior is more delicate, but as mentioned in our previous work ([1] (65)), we can compare with the similar one of (24):

$$I\left(\sigma \right)\sim {(1-{\sigma }^{1/\beta })}^{\eta -1}\mathrm{with}\eta -1=\alpha \gamma -1>-1,as\sigma \to 1,\sigma <1.$$

This generalized Erdélyi-Kober integral preserves the power functions up to a scalar multiplier, similarly to (20), namely,

$${\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }\left\{z^p\right\}={\Theta }_p{z}^p,\mathrm{with}{\Theta }_p={\displaystyle \frac{{\Gamma }^{\gamma }(\nu +1+p\beta )}{{\Gamma }^{\gamma }(\nu +1+p\beta +\alpha )}},p>-{\displaystyle \frac{\nu +1}{\beta }},$$

(27)

and $\alpha \ge 0$, $\nu >-p\beta -1$ are, respectively, the conditions for the operator’s parameters.

To derive (27), we have used the auxiliary result (63) from our recent survey [1], being an analogue of the corresponding integral of an $H_{m,m}^{m,0}$-function from ([10], [App.E, (E.21)]):

$$\begin{gathered} \underset{0}{\overset{1}{\int }}{I}_{m,m}^{m,0}\left[\sigma \left|\begin{array}{c}{(a_i,A_i,{\alpha }_i)}_1^m\\ {(b_i,B_i,{\beta }_i)}_1^m\end{array}\right.\right]d\sigma =\underset{0}{\overset{\infty }{\int }}\dots \mathrm{of} \mathrm{same} I-f.\dots d\sigma \\ =\prod _{i=1}^m{\displaystyle \frac{{\Gamma }^{{\beta }_i}(b_i+B_i)}{{\Gamma }^{{\alpha }_i}(a_i+A_i)}},\forall a_i>b_i>0, \end{gathered}$$

(28)

now for $m=1$:

$$\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha +p\beta ,\beta ,\gamma \\ \nu +1-\beta +p\beta ,\beta ,\gamma \end{array}\right.\right]d\sigma ={\displaystyle \frac{{\Gamma }^{\gamma }(\nu +1+p\beta )}{{\Gamma }^{\gamma }(\nu +1+p\beta +\alpha )}}.$$

(29)

It is easy to present the Mellin transform of the operator (25). We have

$$\mathfrak{M}\{{\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }f\left(z\right);s\}=\underset{0}{\overset{\infty }{\int }}{z}^{s-1}ds\left\{\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(z\sigma \right)d\sigma \right\}$$

$$=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]d\sigma \left\{\underset{0}{\overset{\infty }{\int }}{z}^{s-1}f\left(z\sigma \right)ds\right\}$$

$$=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]\mathfrak{M}\{f(\sigma ·z;s\}d\sigma$$

$$=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]{\sigma }^{-s}\mathfrak{M}\left\{f\right(z;s\}d\sigma$$

$$=\mathfrak{M}\{f(z;s\}·\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha -s\beta ,\beta ,\gamma \\ \nu +1-\beta -s\beta ,\beta ,\gamma \end{array}\right.\right]{\sigma }^{-s}d\sigma ,$$

where we have applied a well-known property of the Mellin transform as $\mathfrak{M}\{f(\sigma ·z);s\}={\sigma }^{s-1}·\mathfrak{M}\{f\left(z\right);s\}$, also (7.3) from [16] and (29). Therefore,

$$\mathfrak{M}\left\{{\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }f\left(z\right);s\right\}=\mathfrak{M}\left\{f(z;s\right\}·{\displaystyle \frac{{\Gamma }^{\gamma }(\nu +1-s\beta )}{{\Gamma }^{\gamma }(\nu +1+\alpha -s\beta )}}.$$

(30)

Then, as a reasoning of the default assumption that ${\mathbb{I}}_{\beta ,\gamma }^{\nu ,0}$ is set to be the identity operator for $\alpha =0$, one can use either the representation (27) that ${\mathbb{I}}_{\beta ,\gamma }^{\nu ,\mathbf{0}}\left\{z^p\right\}=z^p$, or the above relation (30) that $\mathfrak{M}\{{\mathbb{I}}_{\beta ,\gamma }^{\nu ,\mathbf{0}}f\left(z\right);s\}=\mathfrak{M}\{f\left(z\right);s\}$.

Let us discuss the properties of the introduced generalized Erdélyi-Kober integral (25) as an operator of fractional calculus, i.e., as an integral operator of fractional order $\alpha \ge 0$.

It is easy to check its bilinearity and also the commutativity of two different generalized E-K operators, using the properties of the I-function, for example, the symmetry in the sets of parameters’ triplets; see ([16] [Sect.7]).

Lemma 1.

The Semigroup property (Law of indices) for the generalized E-K integral operator holds:

$${\mathbb{I}}_{\beta ,\gamma }^{\nu +{\alpha }_1,{\alpha }_2}{\mathbb{I}}_{\beta ,\gamma }^{\nu ,{\alpha }_1}f\left(z\right)={\mathbb{I}}_{\beta ,\gamma }^{\nu ,{\alpha }_1}{\mathbb{I}}_{\beta ,\gamma }^{\nu +{\alpha }_1,{\alpha }_2}f\left(z\right)={\mathbb{I}}_{\beta ,\gamma }^{\nu ,{\alpha }_1+{\alpha }_2}f\left(z\right),{\alpha }_1\ge 0,{\alpha }_2\ge 0.$$

(31)

Proof of Lemma 1.

For brevity, we introduce the notations

$${}^2\mathbb{I}F\left(z\right)={\mathbb{I}}_{\beta ,\gamma }^{\nu +{\alpha }_1,{\alpha }_2}F\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\tau \left|\begin{array}{c}\nu +1+{\alpha }_1-\beta +{\alpha }_2,\beta ,\gamma \\ \nu +1+{\alpha }_1-\beta ,\beta ,\gamma \end{array}\right.\right]F\left(z\tau \right)d\tau$$

and

$$F\left(z\tau \right)={}^1\mathbb{I}f\left(z\tau \right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(z\tau \sigma \right)d\sigma .$$

Then,

$${}^2\mathbb{I}{}^1\mathbb{I}f\left(z\right)={\mathbb{I}}_{\beta ,\gamma }^{\nu +{\alpha }_1,{\alpha }_2}{\mathbb{I}}_{\beta ,\gamma }^{\nu ,{\alpha }_1}f\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\tau \left|\begin{array}{c}\nu +1+{\alpha }_1-\beta +{\alpha }_2,\beta ,\gamma \\ \nu +1+{\alpha }_1-\beta ,\beta ,\gamma \end{array}\right.\right]d\tau$$

$$\times \left\{\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(z\tau \sigma \right)d\sigma \right\}$$

$$=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]d\sigma$$

$$\times \left\{\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\tau \left|\begin{array}{c}\nu +1+{\alpha }_1-\beta +{\alpha }_2,\beta ,\gamma \\ \nu +1+{\alpha }_1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(z\tau \sigma \right)d\tau \right\},$$

interchanging the order of integrations. Let us note the limits in which the variables change, ${\tau |}_0^1$ and ${\sigma |}_0^1$, and substitute ${y:=\tau \sigma |}_0^1$ in the inner integral with $\tau ={\sigma }^{-1}y,d\tau ={\sigma }^{-1}dy.$ Then,

$${}^2\mathbb{I}{}^1\mathbb{I}f\left(z\right)=\underset{0}{\overset{1}{\int }}{\sigma }^{-1}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]d\sigma$$

$$\times \left\{\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[{\sigma }^{-1}y\left|\begin{array}{c}\nu +1+{\alpha }_1-\beta +{\alpha }_2,\beta ,\gamma \\ \nu +1+{\alpha }_1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(zy\right)dy\right\}$$

$$=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1-\beta ,\beta ,\gamma \\ \nu +1-\beta -\beta ,\beta ,\gamma \end{array}\right.\right]d\sigma$$

$$\times \left\{\underset{0}{\overset{1}{\int }}{I}_{1,1}^{0,1}\left[y^{-1}\sigma \left|\begin{array}{c}-\nu -{\alpha }_1+\beta ,\beta ,\gamma \\ -\nu -{\alpha }_1+\beta -{\alpha }_2,\beta ,\gamma \end{array}\right.\right]f\left(zy\right)dy\right\}$$

$$:=\underset{0}{\overset{1}{\int }}f\left(zy\right)dy\left\{\tilde{\mathbb{I}}\right\},\mathrm{where} \mathrm{we} \mathrm{denote}:$$

$$\tilde{\mathbb{I}}=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1-\beta ,\beta ,\gamma \\ \nu +1-\beta -\beta ,\beta ,\gamma \end{array}\right.\right]I_{1,1}^{0,1}\left[y^{-1}\sigma \left|\begin{array}{c}-\nu +\beta -{\alpha }_1,\beta ,\gamma \\ -\nu +\beta -{\alpha }_1-{\alpha }_2,\beta ,\gamma \end{array}\right.\right]d\sigma .$$

This is after interchanging the order of integrals again, and we use two properties of the I-functions that are very similar to those for the H-functions, namely, (7.3) and (7.5) from Rathie [16]: to enter the multiplier ${\sigma }^{-1}$ inside the $I_{1,1}^{1,0}$-function in the first integral, and to replace $I_{1,1}^{1,0}$ by $I_{1,1}^{0,1}$ of reciprocal argument in the second one. Now, $\tilde{\mathbb{I}}$ as an integral of the product of two different I-functions remains to be evaluated. In general, such a result comes from the evaluation of the Mellin transform of such a product, as in Vellaisamy–Kataria ([31], [Prop.3.1, p.288]). However, in our work [1], we have rewritten this in a simpler form as Lemma 1, Equation (67), for $\lambda \ne 0,\nu \ne 0$, $s>0$:

$$\begin{gathered} \underset{0}{\overset{\infty }{\int }}{\sigma }^{s-1}·I_{p,q}^{m,n}\left[\lambda \sigma \left|\begin{array}{c}{(a_i,A_i,{\alpha }_i)}_1^p\\ {(b_j,B_j,{\beta }_j)}_1^q\end{array}\right.\right]·I_{u,v}^{k,l}\left[\nu \sigma \left|\begin{array}{c}{(c_i,C_i,{\phi }_i)}_1^u\\ {(d_j,D_j,{\psi }_j)}_1^v\end{array}\right.\right]d\sigma \\ ={\displaystyle \frac{1}{s}}·I_{q+u,p+v}^{n+k,m+l}\left[{\displaystyle \frac{\nu }{\lambda }}\left|\begin{array}{c}{(c_i,C_i,{\phi }_i)}_1^l,{(1-b_j-B_j,B_j,{\beta }_j)}_1^q,{(c_i,C_i,{\phi }_i)}_{l+1}^u\\ {(d_j,D_i,{\psi }_j)}_1^k,{(1-a_i-A_i,A_i,{\alpha }_i)}_1^p,{(d_j,D_j,{\psi }_j)}_{k+1}^v\end{array}\right.\right]. \end{gathered}$$

(32)

We take into consideration that the first involved $I_{1,1}^{1,0}$-function vanishes identically for the argument $\sigma >1$, and the second one $I_{1,1}^{0,1}$ also vanishes for $\sigma >1$, since its argument is $y^{-1}\sigma =\tau <1$. Then, the integral $\underset{0}{\overset{\infty }{\int }}$ in (32) reduces to $\underset{0}{\overset{1}{\int }}$, and we have

$$\tilde{\mathbb{I}}=\underset{0}{\overset{\infty }{\int }}\dots d\sigma =\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +{\alpha }_1-\beta ,\beta ,\gamma \\ \nu +1-\beta -\beta ,\beta ,\gamma \end{array}\right.\right]$$

$$\times I_{1,1}^{0,1}\left[y^{-1}\sigma \left|\begin{array}{c}-\nu +\beta -{\alpha }_1,\beta ,\gamma \\ -\nu +\beta -{\alpha }_1-{\alpha }_2,\beta ,\gamma \end{array}\right.\right]d\sigma$$

$$=I_{2,2}^{0,2}\left[y^{-1}\left|\begin{array}{c}(-\nu +\beta -{\alpha }_1,\beta ,\gamma ),(-\nu +\beta +\beta -\beta ,\beta ,\gamma )\\ (-\nu +\beta -{\alpha }_1-{\alpha }_2,\beta ,\gamma ),(-\nu +\beta +\beta -{\alpha }_1-\beta ,\beta ,\gamma )\end{array}\right.\right]$$

$$=I_{1,1}^{0,1}\left[y^{-1}\left|\begin{array}{c}-\nu +\beta ,\beta ,\gamma \\ -\nu +\beta -{\alpha }_1-{\alpha }_2,\beta ,\gamma \end{array}\right.\right]=I_{1,1}^{1,0}\left[y\left|\begin{array}{c}\nu +1-\beta +{\alpha }_1+{\alpha }_2,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right].$$

The above reduction of $I_{2,2}^{0,2}$ to $I_{1,1}^{0,1}$ is because the triplets $(-\nu +\beta -{\alpha }_1,\beta ,\gamma )$ and $(-\nu +\beta +\beta -{\alpha }_1-\beta ,\beta ,\gamma )=(-\nu +\beta -{\alpha }_1,\beta ,\gamma )$ in the up and bottom parameters annulate each other and thus reduce the orders of the I-function (according to another property (7.1), [16], analogous to that for H-function). The result for $\tilde{\mathbb{I}}$, after again using the mentioned property (7.5), is exactly the kernel function of the operator in r.h.s. of (31),

$${\mathbb{I}}_{\beta ,\gamma }^{\nu ,{\alpha }_1+{\alpha }_2}f\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[y\left|\begin{array}{c}\nu +1-\beta +{\alpha }_1+{\alpha }_2,\beta ,\gamma \\ \nu +1-\beta ,\beta ,\gamma \end{array}\right.\right]f\left(zy\right)dy,$$

which is the desired result. □

Bearing in mind the established operational properties of the introduced generalized Erdélyi-Kober integral (25), based on the I-function theory from [16], we can assure that the axioms for an operator to be an operator of fractional calculus are well satisfied. Therefore, we can consider (25) as an operator of integration of fractional order $\alpha \ge 0$.

4. Particular Cases

  • As already noted, when the fourth parameter $\gamma >0$ is taken as $\gamma =1$, the kernel $I_{1,1}^{1,0}$ function of the generalized Erdélyi-Kober operator (25) reduces to a $H_{1,1}^{1,0}$ function and also to a $G_{1,1}^{1,0}$-function. Specifically, it is the following elementary function:

$$I_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu +1-\beta +\alpha ,\beta ,1\\ \nu +1-\beta ,\beta ,1\end{array}\right.\right]={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{(1-{\sigma }^{\frac{1}{\beta }})}^{\alpha -1}{\sigma }^{\frac{\nu +1}{\beta }-1},$$

and we have the “classical” Erdélyi-Kober integral of fractional order α; see (26):

$${\mathbb{I}}_{\beta ,1}^{\nu ,\alpha }f\left(z\right)=I_{1/\beta }^{\nu ,\alpha }f\left(z\right).$$

Additionally, for $\beta =1$,

$${\mathbb{I}}_{1,1}^{\nu ,\alpha }f\left(z\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{(1-\sigma )}^{\alpha -1}{\sigma }^{\nu }f\left(z\sigma \right)d\sigma ,$$

it is a variant of the Riemann-Liuoville fractional integral but with a “weight” ν, as introduced by Kober, while for $\beta =1/2$ (${\beta }^{\ast }=2)$, this is (up to a multiplier) the Erdélyi-Kober integral as extended and used by Sneddon [26]:

$${\mathbb{I}}_{2,1}^{\nu ,\alpha }f\left(z\right)={\displaystyle \frac{2}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{(1-{\sigma }^2)}^{\alpha -1}{\sigma }^{2\nu -1}f\left(z\sigma \right)d\sigma$$

$$=2z^{2-2\alpha -2\nu }{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{z}{\int }}{(z^2-{\tau }^2)}^{\alpha -1}{\tau }^{2\nu -1}f\left(\tau \right)d\tau ,$$

and later with an arbitrary $\beta >0$.

  • We can introduce a Generalized Riemann-Liouville (R-L) fractional integral of order $\alpha >0$ with an additional fractional index $\gamma >0$. For example, we consider the particular case with $\nu =0$, $\beta =1$:

$${\mathbb{I}}_{1,\gamma }^{0,\alpha }f\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\alpha ,1,\gamma \\ 0,1,\gamma \end{array}\right.\right]f\left(z\sigma \right)d\sigma .$$

(33)

This will be a generalized Riemann-Liouville fractional integral of order $\alpha >0$. For $\alpha =0$, ${\mathbb{I}}_{1,\gamma }^{0,0}f\left(z\right):=f\left(z\right)$ is the identity.

  • In our previous works, we provided a long list of examples of particular cases of Erdélyi-Kober operators that were studied by different authors (operators of Hardy–Littlewood, Uspensky, Sonine, etc.) and employed in various problems and areas, including in Geometric Function Theory (GFT), the theory of the Gelfond-Leontiev-type operators for which the Mittag-Leffler-type and Mittag-Leffler–Le Roy-type functions appear as eigenfunctions, etc.

Indeed, in ([10], [Ch.2]) (also see details in [1]), we considered the so-called Gelfond-Leontiev (G-L) generalized integration and differentiation operators generated by an entire function $\phi \left(\lambda \right)$, [32]. The core of the G-L theory is that to an analytic function $f\left(z\right)$ defined by means of a power series; we present its images constructed as Hadamard products by quotients of “shifted” coefficients of $\phi \left(\lambda \right)$. In a particular case, we took $\phi$ the Mittag-Leffler function as an entire function

$$\phi \left(\lambda \right):=E_{\alpha ,\nu }\left(\lambda \right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k}{\Gamma (\alpha k+\nu )}},\alpha >0,\nu \in \mathbb{R},$$

and constructed the corresponding G-L operators of generalized integration $l_{\alpha ,\nu }$, resp. of generalized differentiation $d_{\alpha ,\nu }$. It has been proven that the power series expressions for these operators can also be presented by means of an Erdélyi-Kober fractional integral:

$$l_{\alpha ,\nu }f\left(z\right)={\displaystyle \frac{z}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{1}{\int }}{(1-\sigma )}^{\alpha -1}{\sigma }^{\nu -1}f\left(z{\sigma }^{\alpha }\right)d\sigma =z{I}_{1/\alpha }^{\nu -1,\alpha }f\left(z\right),$$

(34)

and as a corresponding Erdélyi-Kober fractional derivative

$$d_{\alpha ,\nu }f\left(z\right)=D_{1/\alpha }^{\nu -1,\alpha }{z}^{-1}f\left(z\right).$$

The following relations have been proven:

$$\begin{array}{ccc}\hfill l_{\alpha ,\nu }{E}_{\alpha ,\nu }\left(\lambda z\right)& =& {\displaystyle \frac{1}{\lambda }}{E}_{\alpha ,\nu }\left(\lambda z\right)-{\displaystyle \frac{1}{\lambda \Gamma \left(\nu \right)}},\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill d_{\alpha ,\nu }{E}_{\alpha ,\nu }\left(\lambda z\right)& =& \lambda E_{\alpha ,\nu }\left(\lambda z\right),\lambda \ne 0,\hfill \end{array}$$

(36)

that is, the Mittag-Leffler function $E_{\alpha ,\nu }\left(z\right)$ is an eigenfunction for the above Erdélyi-Kober operators.

A similar situation arises with the generalized Erdélyi-Kober integral (25). Recently, an interest has raised in the not-so-popular Le Roy function $F^{\left(\gamma \right)}\left(z\right)$, introduced in [33], almost the same time as the Mittag-Leffler function in [34], and for rather similar purposes not related to fractional calculus at all. Several authors have considered a Mittag-Leffler-type analogue $F_{\alpha ,\nu }^{\left(\gamma \right)}$ of the Le Roy function, called M-L type Le Roy (MLR) function (for example, Garrappa–Rogosin–Mainardi [35]):

$$F^{\left(\gamma \right)}\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{\left[(k!)\right]}^{\gamma }}},F_{\alpha ,\nu }^{\left(\gamma \right)}\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{[\Gamma (\alpha k+\nu )]}^{\gamma }}},\forall \alpha ,\nu ,\gamma >0.$$

(37)

This MLR function is an entire function of order $1/\alpha \gamma$, and we have represented it also in terms of the Rathie I-function (see [1]):

$$F_{\alpha ,\nu }^{\left(\gamma \right)}\left(z\right)=I_{1,2}^{1,1}\left[-z\left|\begin{array}{c}(0,1,1)\\ (0,1,1),(1-\nu ,\alpha ,\gamma )\end{array}\right.\right].$$

In our work [1], we considered the problem of finding suitable integral and differential operators for which the Le Roy-type functions (37) are eigenfunctions. Then, one can expect in what kind of integral/differential equations these special functions would appear as solutions.

Again, we used the apparatus of the Gelfond-Leontiev operators, this time generated by the Le Roy-type functions. For the multi-index extensions ($m=1,2,3,\dots$) of (37), the G-L operators are presented by (57)–(58) in [1], by means of power series, and our Theorem 5 there states that these are their eigen operators. In the case $m=1$ related to $F_{\alpha ,\nu }^{\left(\gamma \right)}\left(z\right)$, the G-L integration operator $L_{MLR}$ can be written as the integral operator

$$Lf\left(z\right):=L_{MLR}f\left(z\right)=\underset{0}{\overset{1}{\int }}{I}_{1,1}^{1,0}\left[\sigma \left|\begin{array}{c}\nu ,\alpha ,\gamma \\ \nu -\alpha ,\alpha ,\gamma \end{array}\right.\right]f\left(z\sigma \right)d\sigma .$$

(38)

We have proven the following “eigen” relations:

$$L{F}_{\alpha ,\nu }^{\left(\gamma \right)}\left(\lambda z\right)={\displaystyle \frac{1}{\lambda }}{F}_{\alpha ,\nu }^{\left(\gamma \right)}\left(\lambda z\right)-{\displaystyle \frac{1}{\lambda {\Gamma }^{\gamma }\left(\nu \right)}},D{F}_{\alpha ,\nu }^{\left(\gamma \right)}\left(\lambda z\right)=\lambda F_{\alpha ,\nu }^{\left(\gamma \right)}\left(\lambda z\right),\lambda \ne 0,$$

where $D:=D_{MLR}$ denotes the corresponding G-L differentiation operator. It is now observed that the “eigen” integral operator (38) for $F_{\alpha ,\nu }^{\left(\gamma \right)}$ is a case of the introduced generalized Erdélyi-Kober fractional integral (25), namely,

$$L_{MLR}f\left(z\right)={\mathbb{I}}_{\alpha ,\gamma }^{\nu +\beta -\alpha ,\alpha }f\left(z\right),\alpha >0,\gamma >0,\nu \in R,$$

and also an example of application of the Gelfond-Leontiev theory [32].

5. New Results: Images of Special Functions Under the Generalized Erdélyi-Kober Fractional Integral

First, let us present our general formula for the image of an arbitrary I-function, under the generalized Erdélyi-Kober fractional integral.

Theorem 1.

Consider the generalized Erdélyi-Kober fractional integral (25) of an arbitrary I-function (9). The result is again an I-function but three of its orders are increased by 1 and the parameters of the fractional integral take part as additional components:

$$\begin{gathered} {\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }\left\{I_{p,q}^{m,n}\left[z\left|\begin{array}{c}{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q\end{array}\right.\right]\right\} \\ =I_{p+1,q+1}^{m,n+1}\left[z\left|\begin{array}{c}(-\nu ,\beta ,\gamma ),{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q,(-\nu -\alpha ,\beta ,\gamma )\end{array}\right.\right]. \end{gathered}$$

(39)

Proof of Theorem 1.

The proof may be presented in a similar way to that in Lemma 1, by using the Formula (32) for the integral of the product of two I-functions, and the operational properties of the Rathie I-function from [16].

Alternatively, we can also process the proof by using the method of Kataria and Vellaisamy [36], applying the result for the image of a power function $z^p$ under the (classical) E-K integral $I_1^{\nu ,\alpha }$. But in the case of the generalized E-K integral ${\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }$, we use the Formula (27). Then, we replace the function $I_{p,q}^{m,n}$ with its Mellin–Barnes integral definition (9) and interchange the orders of integrals. □

Let us note that in their paper [36], the authors start by treating first the more complicated case of the Marichev-Saigo-Maeda (M-S-M) operators of FC, then the simpler Saigo fractional integral, and finally the simplest case of Erdélyi-Kober (E-K) integral, as a corollary. However, ours are the only works (see [10,24,28], etc.) in which the fact that the Saigo hypergeometric operator is just a composition of two ($m=2$) E-K integrals and the M-S-M operator is a composition of three ($m=3$) E-K integrals has been observed and used, all these as particular cases of the Generalized Fractional Calculus operators [10], based on commutable compositions of $m=1,2,3,\dots ,$ E-K integrals. Therefore, it is enough to evaluate only the E-K image of some special function and then apply the result subsequently m times.

Note that for the (classical) Erdélyi-Kober fractional integral of an arbitrary I-function, Corollary 3.3 in Kataria and Vellaisamy [36] agrees well with a particular case of our Formula (39); namely, for $\rho =1$, $a=1$, $\mu =1$ there, the result reads as

$$\begin{gathered} I_1^{\nu ,\alpha }\left\{I_{p,q}^{m,n}\left[z\left|\begin{array}{c}{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q\end{array}\right.\right]\right\} \\ =I_{p+1,q+1}^{m,n+1}\left[z\left|\begin{array}{c}(-\nu ,1,1),{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q,(-\nu -\alpha ,1,1)\end{array}\right.\right]. \end{gathered}$$

(40)

Also, Srivastava et al. ([20], [(2.4)]) presented a Riemann–Lioville integral (17), (22) of the Inayat–Hussain $\overline{H}$-function, as a simpler variant of the I-function. In our denotations, their result reads as follows:

$$I^{\alpha }\left\{z^{\lambda -1}{\overline{H}}_{p,q}^{m,n}\left(\omega z^{\kappa }\right)\right\}=z^{\lambda +\alpha -1}{\overline{H}}_{p+1,q+1}^{m,n+1}\left[\omega z^{\kappa }\left|\begin{array}{c}(1-\lambda ,\kappa ,1),{(a_j,A_j,{\alpha }_j)}_1^n,{(a_j,A_j,1)}_{n+1}^p\\ {(b_k,B_k,1)}_1^m,{(b_k,B_k,{\beta }_k)}_{m+1}^q,(1-\lambda -\alpha ,\kappa ,1)\end{array}\right.\right],$$

as a particular case of our result in the above Theorem 1.

We can immediately draw a parallel between the above Theorem 1, and the corresponding result for the “classical” E-K integral of an arbitrary H-function; see ([28], [Theorem 2]) (like in (18), we set ${\beta }^{\ast }=1/\beta )$:

$$\begin{gathered} I_{{\beta }^{\ast }}^{\nu ,\alpha }\left\{H_{p,q}^{m,n}\left[z\left|\begin{array}{c}{(a_j,A_j)}_1^p\\ {(b_k,B_k)}_1^q\end{array}\right.\right]\right\} \\ =H_{p+1,q+1}^{m,n+1}\left[z\left|\begin{array}{c}{(a_j,A_j)}_1^n,(-\nu ,{\beta }^{\ast }),{(a_j,A_j)}_{n+1}^p\\ {(b_k,B_k)}_1^m,(-\nu -\alpha ,{\beta }^{\ast }),{(b_k,B_k)}_{m+1}^q\end{array}\right.\right]. \end{gathered}$$

The most useful and expected result will be for the image of the generalized Fox–Wright function (13), because many of the special functions that are not cases of the Fox H-function are practically ${}_p\tilde{\Psi }_q$-functions, by analogy with the SF of FC from [12] that are (simpler) ${}_p\Psi _q$-functions.

Theorem 2.

Assume the same conditions for the parameters of the generalized E-K operator (25) and of the ${}_p\tilde{\Psi }_q$-function (13). The following image formula holds:

$$\begin{gathered} {\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }\left\{{}_p\tilde{\Psi }_q\left[\left.\begin{array}{c}{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q\end{array}\right|z\right]\right\} \\ ={}_{p+1}\tilde{\Psi }_{q+1}\left[\left.\begin{array}{c}(\nu +1,\beta ,\gamma ),{(a_j,A_j,{\alpha }_j)}_1^p\\ {(b_k,B_k,{\beta }_k)}_1^q,(\nu +1+\alpha ,\beta ,\gamma )\end{array}\right|z\right]. \end{gathered}$$

(41)

Proof of Theorem 2.

This is a corollary of Theorem 1, where we have to use the representation (16) of the ${}_p\tilde{\Psi }_q$-function as an I-function. □

Again, observe that a parallel with the image under a “classical” E-K integral of a Fox–Wright function ${}_p\Psi _q$ holds, ([12], [Lemma 1]), namely,

$$\begin{gathered} I_{{\beta }^{\ast }}^{\nu ,\alpha }\left\{{}_p\Psi _q\left[\left.\begin{array}{c}{(a_j,A_j)}_1^p\\ {(b_k,B_k)}_1^q\end{array}\right|z\right]\right\} \\ ={}_{p+1}\Psi _{q+1}\left[\left.\begin{array}{c}(\nu +1,{\beta }^{\ast }),{(a_j,A_j)}_1^p\\ {(b_k,B_k)}_1^q,(\nu +1+\alpha ,{\beta }^{\ast })\end{array}\right|z\right]. \end{gathered}$$

One can also compare the image of the ${}_p\tilde{\Psi }_q$ under the E-K operator (18), ([2], [Theorem 4]). We have

$$\begin{array}{l}{I}_{{\beta }^{\ast }}^{\nu ,\alpha }\left\{z^c{}_p\tilde{\Psi }_q\left[\left.\begin{array}{c}{(a_j,A_j,{\alpha }_j)}_{j=1}^p\\ {(b_k,B_k,{\beta }_k)}_{k=1}^q\end{array}\right|\lambda z^{\omega }\right]\right\}\\ =z^c{}_{p+1}\tilde{\Psi }_{q+1}\left[\left.\begin{array}{c}(\nu +1+c{\beta }^{\ast },\omega {\beta }^{\ast },1),{(a_j,A_j;{\alpha }_j)}_{j=1}^p\\ {(b_k,B_k,{\beta }_k)}_{k=1}^q,(\nu +1+\alpha +c{\beta }^{\ast },\omega {\beta }^{\ast },1)\end{array}\right|\lambda z^{\omega }\right].\end{array}$$

(42)

Images of some particular cases of ${}_p\tilde{\Psi }_q$and of I-function

• Le Roy–Mittag-Leffler function (37) (here, we adapted the denotation of parameters). According to ([2], [eq. (57)]),

$$F_{\mu ,\lambda }^{\eta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{\Gamma }^{\eta }(\mu k+\lambda )}}={}_1\tilde{\Psi }_1\left[\left.\begin{array}{c}(1,1,1)\\ (\lambda ,\mu ,\eta )\end{array}\right|z\right]$$

$$=I_{1,2}^{1,1}\left[-z\left|\begin{array}{c}(0,1,1)\\ (0,1,1),(1-\lambda ,\mu ,\eta )\end{array}\right.\right].$$

Then, from (41), we have

$$\begin{gathered} {\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }\left\{F_{\mu ,\lambda }^{\eta }\left(z\right)\right\}={\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }\left\{{}_1\tilde{\Psi }_1\left[\left.\begin{array}{c}(1,1,1)\\ (\lambda ,\mu ,\eta )\end{array}\right|z\right]\right\} \\ ={}_2\tilde{\Psi }_2\left[\left.\begin{array}{c}(\nu +1,\beta ,\gamma ),(1,1,1)\\ (\lambda ,\mu ,\eta ),(\nu +1+\alpha ,\beta ,\gamma )\end{array}\right|z\right]. \end{gathered}$$

(43)

When $\eta =1$, this gives the image of the Mittag-Leffler function $E_{\mu ,\lambda }$, and, respectively, when $\eta >0$, $\mu =1, \lambda =0$, the image of the original Le Roy function [33], denoted below by

$$E_{\mu ,\lambda }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\mu k+\lambda )}},\mathrm{resp}.F^{\eta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{\left[(k!)\right]}^{\eta }}}.$$

• Polylogarithm function (see, e.g., in ([3], [Vol.1]), [22]). According to ([2], [eq. (62)]),

$${\mathrm{Li}}_{\eta }\left(z\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{z^k}{k^{\eta }}}=\sum _{k=1}^{\infty }{\displaystyle \frac{{\Gamma }^{\eta }\left(k\right)}{{\Gamma }^{\eta }(k+1)}}{z}^k={}_2\tilde{\Psi }_1\left[\left.\begin{array}{c}(0,1,\eta ),(1,1,1)\\ (1,1,\eta )\end{array}\right|z\right],$$

is an analytic function for $\left|z\right|<1$, while for $\left|z\right|=1$, the series is convergent for $Re\eta >1$. Observe the particular cases ${\mathrm{Li}}_1\left(z\right)=-\ln (1-z)$; ${\mathrm{Li}}_0\left(z\right)={\displaystyle \frac{z}{1-z}}$; ${\mathrm{Li}}_{\eta }\left(1\right)=\zeta \left(\eta \right)={\displaystyle \sum _{k=1}^{\infty }\frac{1}{k^{\eta }}}$, the Riemmann Zeta function.

Then, in view of (41), its generalized Erdélyi-Kober integral image is

$${\mathbb{I}}_{\beta ,\gamma }^{\nu ,\alpha }\left\{{\mathrm{Li}}_{\eta }\left(z\right)\right\}={}_3\tilde{\Psi }_2\left[\left.\begin{array}{c}(n+1,\beta ,\gamma ),(0,1,\eta ),(1,1,1)\\ (1,1,\eta ),(\nu +1+\alpha ,\beta ,\gamma )\end{array}\right|z\right].$$

(44)

Similarly, applying Theorem 2, the images under the generalized E-K fractional integral can be written for many other special functions like the Riemann Zeta function, Hurwitz–Lerch Zeta function, Feynman integrals (see, e.g., [16,17,18]), etc., that are cases of the ${}_p\tilde{\Psi }_q$-function, as listed in our surveys [1,2], and for their simpler cases of SF of FC, from [12,28].

6. Discussions for Further Research: Compositions of Generalized Erdélyi-Kober  Integrals

There are many FC operators treated in other authors’ works as extensions or cases of the Riemann-Liouville and Erdélyi-Kober fractional integrals. Among them, the most popular are the Saigo operators and the Marichev-Saigo-Maeda (M-S-M) operators. But as mentioned after Theorem 1, these are just compositions of the Erdélyi-Kober fractional integrals; see [10,24,28] and our next joint works.

• For example, the Saigo fractional integral operator has been originally defined with a Gauss hypergeometric function in the kernel, but as we observed, it can also be represented as a commutative composition of two Erdélyi-Kober integrals; see details in [10], and ([24], [(65)–(66)]):

$$\begin{gathered} I^{\alpha ,{\beta }^{\ast },\eta }f\left(z\right)={\displaystyle \frac{z^{-\alpha -{\beta }^{\ast }}}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{z}{\int }}{(z-t)}^{\alpha -1}{}_{2}F_1(\alpha +{\beta }^{\ast },-\eta ,\alpha ;1-{\displaystyle \frac{t}{z}})f\left(t\right)dt \\ =z^{-{\beta }^{\ast }}{I}_1^{0,\alpha +\eta }{I}_1^{\eta -{\beta }^{\ast },-\eta }f\left(z\right)=z^{-{\beta }^{\ast }}{I}_1^{\eta -{\beta }^{\ast },0}{I}_1^{-\eta ,\alpha +\eta }f\left(z\right). \end{gathered}$$

(45)

This is the operator also considered in Kataria et al. [36]; see (7) there. In ([36], [Cor. 3.2]), these authors proposed the image of an arbitrary I-function, and this will be a particular case of such an image under the more general operator of Saigo type that we can now define by adding an arbitrary fractional parameter $\gamma >0$.

Specifically, we introduce the following generalized analogue of the Saigo operator, by a composition of the two generalized Erdélyi-Kober integrals:

$$\begin{gathered} {\mathbb{I}}_{1,\gamma }^{\alpha ,\beta ,\eta }f\left(z\right):=z^{-\beta }{\mathbb{I}}_{1,\gamma }^{0,\alpha +\eta }{\mathbb{I}}_{1,\gamma }^{\eta -\beta ,-\eta }f\left(z\right) \\ =z^{-\beta }\underset{0}{\overset{1}{\int }}{I}_{2,2}^{2,0}\left[\sigma \left|\begin{array}{c}(\alpha ,1,\gamma ),(-\beta ,1,\gamma )\\ (0,1,\gamma ),(\eta -{\beta }^{\ast },1,\gamma )\end{array}\right.\right]f\left(z\sigma \right)d\sigma . \end{gathered}$$

(46)

For $\gamma =1$, it reduces to (45).

• The Marichev-Saigo-Maeda (M-S-M) fractional integral operator has been defined by means of the Appel $F_3$ function (Horn function), with $a,a^{\prime },b,b^{\prime },c$ complex parameters, $\mathrm{Re}c>0$:

$$I^{a,a^{\prime },b,b^{\prime },c}f\left(z\right)={\displaystyle \frac{z^{-a}}{\Gamma \left(c\right)}}\underset{0}{\overset{z}{\int }}{(z-\xi )}^{c-1}{\xi }^{-a^{\prime }}{F}_3(a,a^{\prime },b,b^{\prime };c;1-{\displaystyle \frac{\xi }{z}},1-{\displaystyle \frac{z}{\xi }})f\left(\xi \right)d\xi .$$

(47)

However, the kernel function in (47), defined for $\left|z\right|<1,\left|\xi \right|<1$ by

$$F_3\left(a,a^{\prime },b,b^{\prime },c,z,\xi \right)=\sum _{m,n=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_m{\left(a^{\prime }\right)}_n{\left(b\right)}_m{\left(b^{\prime }\right)}_n}{{\left(c\right)}_{m+n}}}{\displaystyle \frac{z^m{\xi }^n}{m!n!}},$$

can be observed to be a $H_{3,3}^{3,0}$ = simpler $G_{3,3}^{3,0}$ function (see [6] [8.4.51,(2)]), and so, the M-S-M operator appears as a particular case ($m=3$) of the generalized fractional integrals from [10] with $H_{m,m}^{m,0}$- and $G_{m,m}^{m,0}$-kernels, namely,

$$I^{a,a^{\prime },b,b^{\prime },c}f\left(z\right)=z^c{I}_{(1,3)}^{(a,b,c-a^{\prime }-b^{\prime }),(b,c-a^{\prime }{-}^{\prime }b,a^{\prime })}f\left(z\right),$$

these are also compositions of three classical Erdélyi-Kober operators, and we can now extend (47) to a composition of three corresponding Erdélyi-Kober integrals of generalized type (25).

• Going further, we plan to develop a generalized fractional calculus with a kernel being an $I_{m,m}^{m,0}$-function, $m=1,2,3,\dots$, and thus to introduce new fractional integrals of multi-order $({\alpha }_1,\dots ,{\alpha }_m)$, similar to those with $H_{m,m}^{m,0}$-function from Kiryakova [10,24].

Let us remind that the notion “generalized fractional integration operators” was introduced by Kalla [37], who initiated the idea to consider integral operators of the form

$$\mathcal{I}f\left(z\right)=z^{-\gamma -1}\underset{0}{\overset{z}{\int }}\Phi (t/z)t^{\gamma }f\left(t\right)dt=\underset{0}{\overset{1}{\int }}\Phi \left(\sigma \right){\sigma }^{\gamma }f\left(z\sigma \right)d\sigma ,$$

(48)

with an arbitrary kernel-function $\Phi \left(\sigma \right)$ such that the integrals make sense for wide classes of functions $f\left(z\right)$ and can satisfy the axioms of FC. Examples of such FC operators appeared by using the Gauss hypergeometric function and Bessel function and arbitrary G- and H-functions as kernels. But it happens that a complete theory of Generalized Fractional Calculus can only be developed by a suitable choice of singular kernels as $G_{m,m}^{m,0}$, $G_{m+n,m+n}^{m,n}$, $H_{m,m}^{m,0}$, $H_{m+n,m+n}^{m,n}$. So, the next step can be to employ kernels of the form $I_{m,m}^{m,0}$, $I_{m+n,m+n}^{m,n}$.

7. Conclusions

Let us briefly summarize the novelties and contributions in our paper. First, we emphasize again that the Rathie I-functions introduced in 1997 in [16] for the needs of statistical physics, but somehow ignored by the mathematicians, can now play an important role in expanding the family of special functions, called Special Functions of Fractional Calculus, by incorporating also other important mathematical functions.

Then, using the I-functions as singular kernels, we can further develop the theory of the Generalized Fractional Calculus (Kiryakova [10,24]). In particular, here, we introduced generalized Erdélyi-Kober integral operators with $I_{1,1}^{1,0}$-kernels, proved that these can be considered new operators of generalized fractional integration and also proposed the images of wide classes of special functions, in a general scheme.

Last but not least, let us briefly comment on the relations of the theoretically studied operators of the classical and generalized Fractional Calculus (FC) with the fractional order models of phenomena in the real physical and social world. After the initial pessimistic conjectures that FC is an interesting but purely exotic topic, since the end of 20th century, its practical applications have flourished as an unavoidable tool for better and more adequate modeling of the fractal world. We just like to mention some discussions at a round table of an international conference in Bulgaria (1996) about the semantical coincidence of the first letters in “Fractals" and “Fractional Calculus" (we may refer the readers to the notes in its proceedings, “A long standing conjecture failed?”, at https://www.researchgate.net/publication/280153125_A_long_standing_conjecture_failed, accessed on 25 August 2025). Yet before, the editor of the proceedings of another conference on fractional calculus in Tokyo (1989) commented on the paper [38], as one of the first where the notions of “Fractal Hausdorff dimension” and “Fractal geometry” have been exploited together, as follows: “Though the present paper seems to be remotely related to fractional calculus, the editor accepted it to insert in the Proceedings in anticipation of the appearance of certain relations with fractional calculus in the future". Soon came the pioneering works of Podlubny (see, e.g., [39], etc.) on geometrical and physical meanings of the operators of FC and numerous next publications where the classical and multi-indexed (generalized) Erdélyi-Kober operators have been used and practically interpreted; see, for example, Herrmann [40], etc. Nevertheless, the 21st century has seen a remarkable boom (accompanied with a dangerous flood) of publications and editions on the applications of fractional calculus theory in mathematical models of all kinds of phenomena in the fractal world.