Abelian Groups of Fractional Operators ^†

Abstract

Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators.

1. Introduction

Fractional calculus is a branch of mathematics that uses derivatives of non-integer order that originated around the same time as conventional calculus due to Leibniz’s notation for derivatives of integer order

$$\begin{array}{c}\hfill {\displaystyle \frac{d^n}{d{x}^n}}.\end{array}$$

Therefore, thanks to this notation, L’Hopital could ask in a letter to Leibniz about the interpretation of taking $n=1/2$ in a derivative. Since at that moment Leibniz could not give a physical or geometrical interpretation of this question, he simply answered to L’Hopital in a letter, “⋯ is an apparent paradox of which, one day, useful consequences will be drawn” [1]. The name of fractional calculus comes from a historical question since, in this branch of mathematical analysis, the derivatives and integrals of a certain order $\alpha$ are studied, with $\alpha \in \mathbb{R}$. Currently, fractional calculus does not have a unified definition of what is considered a fractional derivative. As a consequence, when it is not necessary to explicitly specify the form of a fractional derivative, it is usually denoted as follows

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{\alpha }}{d{x}^{\alpha }}}.\end{array}$$

The fractional operators have many representations, but one of their fundamental properties is that they allow retrieving the results of conventional calculus when $\alpha \to n$. For example, let $f:\Omega \subset \mathbb{R}\to \mathbb{R}$ be a function such that $f\in L_{loc}^1(a,b)$, where $L_{loc}^1(a,b)$ denotes the space of locally integrable functions on the open interval $(a,b)\subset \Omega$. One of the fundamental operators of fractional calculus is the operator Riemann–Liouville fractional integral, which is defined as follows [2,3]:

$$\begin{array}{c}\hfill {}_a^{}{I}_x^{\alpha }f\left(x\right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^x{(x-t)}^{\alpha -1}f\left(t\right)dt,\end{array}$$

(1)

where $\Gamma$ denotes the Gamma function. It is worth mentioning that the above operator is a fundamental piece to construct the operator Riemann-Liouville fractional derivative, which is defined as follows [2,4]:

$$\begin{array}{c}\hfill \begin{array}{c}{}_a^{}{D}_x^{\alpha }f\left(x\right):=\left\{\begin{array}{cc}{\displaystyle {}_a^{}{I}_x^{-\alpha }f\left(x\right),}& \mathrm{if}\alpha <0\\ {\displaystyle {\displaystyle \frac{d^n}{d{x}^n}}\left({}_a^{}{I}_x^{n-\alpha }f\left(x\right)\right),}& \mathrm{if}\alpha \ge 0\end{array}\right.\end{array},\end{array}$$

(2)

where $n=\lceil \alpha \rceil$ and ${}_a^{}{I}_x^{0}f\left(x\right):=f\left(x\right)$. On the other hand, let $f:\Omega \subset \mathbb{R}\to \mathbb{R}$ be a function n-times differentiable such that $f,f^{\left(n\right)}\in L_{loc}^1(a,b)$. Then, the Riemann–Liouville fractional integral also allows constructing the operator Caputo fractional derivative, which is defined as follows [2,4]:

$$\begin{array}{c}\hfill \begin{array}{c}{}_a^C{D}_x^{\alpha }f\left(x\right):=\left\{\begin{array}{cc}{\displaystyle {}_a^{}{I}_x^{-\alpha }f\left(x\right),}& \mathrm{if}\alpha <0\\ {\displaystyle {}_a^{}{I}_x^{n-\alpha }{f}^{\left(n\right)}\left(x\right),}& \mathrm{if}\alpha \ge 0\end{array}\right.\end{array},\end{array}$$

(3)

where $n=\lceil \alpha \rceil$ and ${}_a^{}{I}_x^0{f}^{\left(n\right)}\left(x\right):=f^{\left(n\right)}\left(x\right)$. Furthermore, if the function f fulfills that $f^{\left(k\right)}\left(a\right)=0\forall k\in \left\{0,1,\cdots ,n-1\right\}$, the Riemann–Liouville fractional derivative coincides with the Caputo fractional derivative, that is,

$$\begin{array}{c}\hfill {}_a^{}{D}_x^{\alpha }f\left(x\right)={}_a^C{D}_x^{\alpha }f\left(x\right).\end{array}$$

(4)

Therefore, applying the operator (2) with $a=0$ to the function $x^{\mu }$, with $\mu >-1$, we obtain the following result:

$$\begin{array}{cc}\hfill {}_0^{}{D}_x^{\alpha }{x}^{\mu }={\displaystyle \frac{\Gamma \left(\mu +1\right)}{\Gamma \left(\mu -\alpha +1\right)}}{x}^{\mu -\alpha },& \alpha \in \mathbb{R}\setminus \mathbb{Z},\end{array}$$

(5)

where if $1\le \lceil \alpha \rceil \le \mu$, it is fulfilled that ${}_0^{}{D}_x^{\alpha }{x}^{\mu }={}_0^C{D}_x^{\alpha }{x}^{\mu }$. To illustrate a bit the diversity of representations that fractional operators may have, we proceed to present a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature [5,6,7]:

• Grünwald-Letnikov fractional derivative:

  $$\begin{array}{cc}\hfill {}_a^{GL}{D}_x^{\alpha }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\alpha }}}\sum _{k=0}^n{\displaystyle \frac{{(-1)}^k\Gamma \left(\alpha +1\right)}{\Gamma \left(k+1\right)\Gamma \left(\alpha -k+1\right)}}f(x-kh),& n=\lfloor (x-a)/h\rfloor .\end{array}$$
• Marchaud fractional derivative:

  $$\begin{array}{cc}\hfill {}_{-\infty }^{Ma}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{\alpha }{\Gamma \left(1-\alpha \right)}}{\int }_{-\infty }^x{(x-t)}^{-\alpha -1}\left(f\left(x\right)-f\left(t\right)\right)dt,& 0<\alpha <1.\end{array}$$
• Hadamard fractional derivative:

  $$\begin{array}{cc}\hfill {}_a^{Ha}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{x}{\Gamma \left(1-\alpha \right)}}{\displaystyle \frac{d}{dx}}{\int }_a^x{\left(ln\left(x\right)-ln\left(t\right)\right)}^{2-\alpha }{\displaystyle \frac{f\left(t\right)}{t}}dt,& 0<\alpha <1.\end{array}$$
• Chen fractional derivative:

  $$\begin{array}{cc}\hfill {}_a^{Ch}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(1-\alpha \right)}}{\displaystyle \frac{d}{dx}}{\int }_a^x{(x-t)}^{-\alpha }f\left(t\right)dt,& 0<\alpha <1.\end{array}$$
• Caputo-Fabrizio fractional derivative:

  $$\begin{array}{c}\hfill \begin{array}{ccc}{\displaystyle {}_a^{CF}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}{\int }_a^{x}exp\left(-\frac{\alpha }{1-\alpha }(x-t)\right)f^{\left(1\right)}\left(t\right)dt,}& 0<\alpha <1,& M\left(0\right)=M\left(1\right)=1.\end{array}\end{array}$$
• Atangana-Baleanu-Caputo fractional derivative:

  $$\begin{array}{c}\hfill \begin{array}{ccc}{\displaystyle {}_a^{ABC}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}{\int }_a^x{E}_{\alpha }\left(-\frac{\alpha }{1-\alpha }{(x-t)}^{\alpha }\right)f^{\left(1\right)}\left(t\right)dt,}& 0<\alpha <1,& M\left(0\right)=M\left(1\right)=1.\end{array}\end{array}$$
• Canavati fractional derivative:

  $$\begin{array}{cc}\hfill {}_a^{Ca}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(1+\alpha -n\right)}}{\displaystyle \frac{d}{dx}}{\int }_a^x{(x-t)}^{n-\alpha }{\displaystyle \frac{d^n}{d{t}^n}}f\left(t\right)dt,& n=\lfloor \alpha \rfloor .\end{array}$$
• Jumarie fractional derivative:

  $$\begin{array}{cc}\hfill {}_a^{Ju}{D}_x^{\alpha }f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(n-\alpha \right)}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_a^x{(x-t)}^{n-\alpha -1}\left(f\left(t\right)-f\left(a\right)\right)dt,& n=\lceil \alpha \rceil .\end{array}$$
• Hadamard fractional integral:

  $$\begin{array}{c}\hfill {}_a^{Ha}{I}_x^{\alpha }f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^x{\left(ln\left(t\right)-ln\left(x\right)\right)}^{\alpha -1}{\displaystyle \frac{f\left(t\right)}{t}}dt.\end{array}$$
• Weyl fractional integral:

  $$\begin{array}{c}\hfill {}_x^{}{W}_{\infty }^{\alpha }f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_x^{\infty }{(t-x)}^{\alpha -1}f\left(t\right)dt.\end{array}$$
• Conformable fractional operator:

  $$\begin{array}{c}\hfill T_{\alpha }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(x+h{x}^{1-\alpha }\right)-f\left(x\right)}{h}}.\end{array}$$
• Katugampola fractional operator:

  $$\begin{array}{c}\hfill D^{\alpha }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(xexp\left(h{x}^{-\alpha }\right)\right)-f\left(x\right)}{h}}.\end{array}$$
• Deformable fractional operator:

  $$\begin{array}{cc}\hfill {\mathcal{D}}^{\alpha }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{(1+h\beta )f(x+h\alpha )-f\left(x\right)}{h}},& \alpha +\beta =1.\end{array}$$

Before continuing, it is worth mentioning that the applications of fractional operators have spread to different fields of science, such as finance [8,9], economics [10,11], number theory through the Riemann zeta function [12,13], in engineering with the study for the manufacture of hybrid solar receivers [14,15], and in physics and mathematics to solve nonlinear algebraic equation systems [16,17,18,19,20,21,22,23,24,25], which is a classical problem in mathematics, physics and engineering that consists of finding the set of zeros of a function $f:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^n$, that is,

$$\begin{array}{c}\hfill \left\{\xi \in \Omega :∥f\left(\xi \right)∥=0\right\},\end{array}$$

where $∥·∥:{\mathbb{R}}^n\to \mathbb{R}$ denotes any vector norm, or equivalently,

$$\begin{array}{c}\hfill \left\{\xi \in \Omega :{\left[f\right]}_k\left(\xi \right)=0\forall k\ge 1\right\},\end{array}$$

where ${\left[f\right]}_k:{\mathbb{R}}^n\to \mathbb{R}$ denotes the k-th component of the function f.

2. Sets of Fractional Operators

Before continuing, it is worth mentioning that due to the large number of fractional operators that exist [5,6,7,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41], it seems that the most natural way to fully characterize the elements of the fractional calculus is by using sets, which is the main idea behind of the methodology known as fractional calculus of sets [42,43]. Therefore, considering a scalar function $h:{\mathbb{R}}^m\to \mathbb{R}$ and the canonical basis of ${\mathbb{R}}^m$ denoted by ${\left\{{\widehat{e}}_k\right\}}_{k\ge 1}$, it is feasible to define the following fractional operator of order $\alpha$ using Einstein’s notation

$$\begin{array}{c}\hfill o_x^{\alpha }h\left(x\right):={\widehat{e}}_k{o}_k^{\alpha }h\left(x\right).\end{array}$$

(6)

Therefore, denoting by ${\partial }_k^n$ the partial derivative of order n applied with respect to the k-th component of the vector x, using the previous operator, it is feasible to define the following set of fractional operators

$$\begin{array}{c}\hfill O_{x,\alpha }^n\left(h\right):=\left\{o_x^{\alpha }:\exists o_k^{\alpha }h\left(x\right)\mathrm{and}\underset{\alpha \to n}{lim}{o}_k^{\alpha }h\left(x\right)={\partial }_k^{n}h\left(x\right)\forall k\ge 1\right\},\end{array}$$

(7)

which corresponds to a nonempty set since it contains the following sets of fractional operators

$$\begin{array}{c}{O}_{0,x,\alpha }^n\left(h\right):=\left\{o_x^{\alpha }:\exists o_k^{\alpha }h\left(x\right)=\left({\partial }_k^n+\mu \left(\alpha \right){\partial }_k^{\alpha }\right)h\left(x\right)\mathrm{and}\underset{\alpha \to n}{lim}\mu \left(\alpha \right){\partial }_k^{\alpha }h\left(x\right)=0\forall k\ge 1\right\}.\end{array}$$

(8)

As a consequence, it is feasible to obtain the following result:

$$\begin{array}{c}\hfill \mathrm{If}{o}_{i,x}^{\alpha },o_{j,x}^{\alpha }\in O_{x,\alpha }^n\left(h\right)\mathrm{with}i\ne j\Rightarrow \exists o_{k,x}^{\alpha }={\displaystyle \frac{1}{2}}\left(o_{i,x}^{\alpha }+o_{j,x}^{\alpha }\right)\in O_{x,\alpha }^n\left(h\right).\end{array}$$

(9)

On the other hand, the complement of the set (7) may be defined as follows

$$\begin{array}{c}\hfill O_{x,\alpha }^{n,c}\left(h\right):=\left\{o_x^{\alpha }:\exists o_k^{\alpha }h\left(x\right)\forall k\ge 1\mathrm{and}\underset{\alpha \to n}{lim}{o}_k^{\alpha }h\left(x\right)\ne {\partial }_k^{n}h\left(x\right)\mathrm{in}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{value}k\ge 1\right\},\end{array}$$

(10)

with which it is feasible to obtain the following result:

$$\begin{array}{c}\hfill \mathrm{If}{o}_{i,x}^{\alpha }={\widehat{e}}_k{o}_{i,k}^{\alpha }\in O_{x,\alpha }^n\left(h\right)\Rightarrow \exists o_{j,x}^{\alpha }={\widehat{e}}_k{o}_{i,{\sigma }_j\left(k\right)}^{\alpha }\in O_{x,\alpha }^{n,c}\left(h\right),\end{array}$$

(11)

where ${\sigma }_j:\left\{1,2,\cdots ,m\right\}\to \left\{1,2,\cdots ,m\right\}$ denotes any permutation different from the identity. Before continuing, it is necessary to mention that set (7) allows generalizing elements of conventional calculus. For example, let ${\mathbb{N}}_0$ be the set $\mathbb{N}\cup \left\{0\right\}$. If $\gamma \in {\mathbb{N}}_0^m$ and $x\in {\mathbb{R}}^m$, then it is feasible to define the following multi-index notation:

$$\left\{\begin{array}{c}\begin{array}{ccc}{\displaystyle \gamma !:=\prod _{k=1}^m{\left[\gamma \right]}_k!,}& {\displaystyle \left|\gamma \right|:=\sum _{k=1}^m{\left[\gamma \right]}_k,}& {\displaystyle x^{\gamma }:=\prod _{k=1}^m{\left[x\right]}_k^{{\left[\gamma \right]}_k}}\end{array}\\ {\displaystyle \frac{{\partial }^{\gamma }}{\partial x^{\gamma }}}:={\displaystyle \frac{{\partial }^{{\left[\gamma \right]}_1}}{\partial {\left[x\right]}_1^{{\left[\gamma \right]}_1}}}{\displaystyle \frac{{\partial }^{{\left[\gamma \right]}_2}}{\partial {\left[x\right]}_2^{{\left[\gamma \right]}_2}}}\cdots {\displaystyle \frac{{\partial }^{{\left[\gamma \right]}_m}}{\partial {\left[x\right]}_m^{{\left[\gamma \right]}_m}}}\end{array}\right..$$

(12)

Therefore, considering a function $h:\Omega \subset {\mathbb{R}}^m\to \mathbb{R}$ and the fractional operator

$$s_x^{\alpha \gamma }\left(o_x^{\alpha }\right):=o_1^{\alpha {\left[\gamma \right]}_1}{o}_2^{\alpha {\left[\gamma \right]}_2}\cdots o_m^{\alpha {\left[\gamma \right]}_m},$$

(13)

it is feasible to define the following set of fractional operators

$$\begin{array}{c}{\displaystyle S_{x,\alpha }^{n,\gamma }\left(h\right):=\left\{s_x^{\alpha \gamma }=s_x^{\alpha \gamma }\left(o_x^{\alpha }\right):\exists s_x^{\alpha \gamma }h\left(x\right)\mathrm{with}{o}_x^{\alpha }\in O_{x,\alpha }^s\left(h\right)\forall s\le n^2\mathrm{and}\underset{\alpha \to k}{lim}{s}_x^{\alpha \gamma }h\left(x\right)={\displaystyle \frac{{\partial }^{k\gamma }}{\partial x^{k\gamma }}}h\left(x\right)\forall \alpha ,\left|\gamma \right|\le n\right\},}\end{array}$$

(14)

from which it is feasible to obtain the following results:

$$\mathrm{If}{s}_x^{\alpha \gamma }\in S_{x,\alpha }^{n,\gamma }\left(h\right)\Rightarrow \left\{\begin{array}{c}{\displaystyle \underset{\alpha \to 0}{lim}{s}_x^{\alpha \gamma }h\left(x\right)=o_1^0{o}_2^0\cdots o_m^{0}h\left(x\right)=h\left(x\right)}\hfill \\ {\displaystyle \underset{\alpha \to 1}{lim}{s}_x^{\alpha \gamma }h\left(x\right)=o_1^{{\left[\gamma \right]}_1}{o}_2^{{\left[\gamma \right]}_2}\cdots o_m^{{\left[\gamma \right]}_m}h\left(x\right)={\displaystyle \frac{{\partial }^{\gamma }}{\partial x^{\gamma }}}h\left(x\right)\forall \left|\gamma \right|\le n}\hfill \\ {\displaystyle \underset{\alpha \to q}{lim}{s}_x^{\alpha \gamma }h\left(x\right)=o_1^{q{\left[\gamma \right]}_1}{o}_2^{q{\left[\gamma \right]}_2}\cdots o_m^{q{\left[\gamma \right]}_m}h\left(x\right)={\displaystyle \frac{{\partial }^{q\gamma }}{\partial x^{q\gamma }}}h\left(x\right)\forall q\left|\gamma \right|\le qn}\hfill \\ {\displaystyle \underset{\alpha \to n}{lim}{s}_x^{\alpha \gamma }h\left(x\right)=o_1^{n{\left[\gamma \right]}_1}{o}_2^{n{\left[\gamma \right]}_2}\cdots o_m^{n{\left[\gamma \right]}_m}h\left(x\right)={\displaystyle \frac{{\partial }^{n\gamma }}{\partial x^{n\gamma }}}h\left(x\right)\forall n\left|\gamma \right|\le n^2}\hfill \end{array}\right..$$

(15)

On the other hand, using little-o notation, it is feasible to obtain the following result:

$$\mathrm{If}x\in B(a;\delta )\Rightarrow \underset{x\to a}{lim}{\displaystyle \frac{o\left({(x-a)}^{\gamma }\right)}{{(x-a)}^{\gamma }}}\to 0\forall \left|\gamma \right|\ge 1,$$

(16)

with which it is feasible to define the following set of functions

$$R_{\alpha \gamma }^n\left(a\right):=\left\{r_{\alpha \gamma }^n:\underset{x\to a}{lim}∥r_{\alpha \gamma }^n\left(x\right)∥=0\forall \left|\gamma \right|\ge n\mathrm{and}∥r_{\alpha \gamma }^n\left(x\right)∥\le o\left({∥x-a∥}^n\right)\forall x\in B(a;\delta )\right\},$$

(17)

where $r_{\alpha \gamma }^n:B(a;\delta )\subset \Omega \to \mathbb{R}$. Therefore, considering the previous set and some $B(a;\delta )\subset \Omega$, it is feasible to define the following sets of fractional operators

$$\begin{array}{c}\begin{array}{c}{\displaystyle T_{x,\alpha ,p}^{n,q,\gamma }(a,h):=\left\{t_x^{\alpha ,p}=t_x^{\alpha ,p}\left(s_x^{\alpha \gamma }\right):s_x^{\alpha \gamma }\in S_{x,\alpha }^{M,\gamma }\left(h\right)\mathrm{and}{t}_x^{\alpha ,p}h\left(x\right):=\sum _{\left|\gamma \right|=0}^p{\displaystyle \frac{1}{\gamma !}}{s}_x^{\alpha \gamma }h\left(a\right){(x-a)}^{\gamma }+r_{\alpha \gamma }^p\left(x\right)\begin{array}{c}\forall \alpha \le n\\ \forall p\le q\end{array}\right\},}\end{array}\end{array}$$

(18)

$$\begin{array}{c}\begin{array}{c}{\displaystyle T_{x,\alpha }^{\infty ,\gamma }(a,h):=\left\{t_x^{\alpha ,\infty }=t_x^{\alpha ,\infty }\left(s_x^{\alpha \gamma }\right):s_x^{\alpha \gamma }\in S_{x,\alpha }^{\infty ,\gamma }\left(h\right)\mathrm{and}{t}_x^{\alpha ,\infty }h\left(x\right):=\sum _{\left|\gamma \right|=0}^{\infty }{\displaystyle \frac{1}{\gamma !}}{s}_x^{\alpha \gamma }h\left(a\right){(x-a)}^{\gamma }\right\},}\end{array}\end{array}$$

(19)

which allow generalizing the Taylor series expansion of a scalar function in multi-index notation [22], where $M=max\left\{n,q\right\}$. As a consequence, it is feasible to obtain the following results:

$$\begin{array}{c}\begin{array}{c}{\displaystyle \mathrm{If}{t}_x^{\alpha ,p}\in T_{x,\alpha ,p}^{1,q,\gamma }(a,h)\mathrm{and}\alpha \to 1\Rightarrow t_x^{1,p}h\left(x\right)=h\left(a\right)+\sum _{\left|\gamma \right|=1}^p{\displaystyle \frac{1}{\gamma !}}{\displaystyle \frac{{\partial }^{\gamma }}{\partial x^{\gamma }}}h\left(a\right){(x-a)}^{\gamma }+r_{\gamma }^p\left(x\right),}\end{array}\end{array}$$

(20)

$$\begin{array}{c}\mathrm{If}{t}_x^{\alpha ,p}\in T_{x,\alpha ,p}^{n,1,\gamma }(a,h)\mathrm{and}p\to 1\Rightarrow t_x^{\alpha ,1}h\left(x\right)=h\left(a\right)+\sum _{k=1}^m{o}_k^{\alpha }h\left(a\right){\left[(x-a)\right]}_k+r_{\alpha \gamma }^1\left(x\right).\end{array}$$

(21)

Finally, it is worth mentioning that the set (7) may be considered as a generating set of sets of fractional tensor operators. For example, considering $\alpha ,n\in {\mathbb{R}}^d$ with $\alpha ={\widehat{e}}_k{\left[\alpha \right]}_k$ and $n={\widehat{e}}_k{\left[n\right]}_k$, it is feasible to define the following set of fractional tensor operators

$$\begin{array}{c}\hfill O_{x,\alpha }^n\left(h\right):=\left\{o_x^{\alpha }:\exists o_x^{\alpha }h\left(x\right)\mathrm{and}{o}_x^{\alpha }\in O_{x,{\left[\alpha \right]}_1}^{{\left[n\right]}_1}\left(h\right)\times O_{x,{\left[\alpha \right]}_2}^{{\left[n\right]}_2}\left(h\right)\times \cdots \times O_{x,{\left[\alpha \right]}_d}^{{\left[n\right]}_d}\left(h\right)\right\}.\end{array}$$

(22)

3. Groups of Fractional Operators

Considering a function $h:\Omega \subset {\mathbb{R}}^m\to {\mathbb{R}}^m$, it is feasible to define sets of fractional operators for a vector function in the following way:

$$\begin{array}{c}\begin{array}{c}{\displaystyle {}_m{O}_{x,\alpha }^n\left(h\right):=\left\{o_x^{\alpha }:o_x^{\alpha }\in O_{x,\alpha }^n\left({\left[h\right]}_k\right)\forall k\le m\right\},}\end{array}\end{array}$$

(23)

$$\begin{array}{c}\begin{array}{c}{\displaystyle {}_m{O}_{x,\alpha }^{n,c}\left(h\right):=\left\{o_x^{\alpha }:o_x^{\alpha }\in O_{x,\alpha }^{n,c}\left({\left[h\right]}_k\right)\forall k\le m\right\},}\end{array}\end{array}$$

(24)

$$\begin{array}{c}{}_m{O}_{x,\alpha }^{n,u}\left(h\right):={}_m{O}_{x,\alpha }^n\left(h\right)\cup {}_m{O}_{x,\alpha }^{n,c}\left(h\right),\end{array}$$

(25)

where ${\left[h\right]}_k:\Omega \subset {\mathbb{R}}^m\to \mathbb{R}$ denotes the k-th component of the function h. Therefore, using the above sets, it is feasible to construct the following family of fractional operators

$$\begin{array}{c}\hfill {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right):=\bigcap _{k\in \mathbb{Z}}{}_m{O}_{x,\alpha }^{k,u}\left(h\right).\end{array}$$

(26)

Before continuing, it should be noted that the above family of fractional operators fulfills the following property with respect to the classical Hadamar product:

$$\begin{array}{c}\hfill o_x^0\circ h\left(x\right):=h\left(x\right)\forall o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right).\end{array}$$

(27)

Furthermore, for each operator $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$, it is feasible to define the following fractional matrix operator [44]:

$$\begin{array}{c}\hfill A_{\alpha }\left(o_x^{\alpha }\right)=\left({\left[A_{\alpha }\left(o_x^{\alpha }\right)\right]}_{jk}\right):=\left(o_k^{\alpha }\right).\end{array}$$

(28)

On the other hand, defining the following modified Hadamard product [42]:

$$\begin{array}{c}\hfill o_{i,x}^{p\alpha }\circ o_{j,x}^{q\alpha }:=\left\{\begin{array}{cc}{o}_{i,x}^{p\alpha }\circ o_{j,x}^{q\alpha },& \mathrm{if}i\ne j\left(\mathrm{Hadamard}\mathrm{product}\mathrm{of}\mathrm{type}\mathrm{horizontal}\right)\hfill \\ o_{i,x}^{(p+q)\alpha },& \mathrm{if}i=j\left(\mathrm{Hadamard}\mathrm{product}\mathrm{of}\mathrm{type}\mathrm{vertical}\right)\hfill \end{array}\right.,\end{array}$$

(29)

for each operator $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$, it is feasible to define an Abelian group of fractional operators isomorphic to the group of integers under the addition, as shown by the following theorem [43,44]:

Theorem 1.

Let $o_x^{\alpha }$ be a fractional operator such that $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$ and let $\left(\mathbb{Z},+\right)$ be the group of integers under the addition. Therefore, considering the modified Hadamard product given by (29), it is feasible to define the following set of fractional matrix operators

$$\begin{array}{c}\hfill {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right):=\left\{A_{\alpha }^{\circ r}=A_{\alpha }\left(o_x^{r\alpha }\right):r\in \mathbb{Z}and{A}_{\alpha }^{\circ r}=\left({\left[A_{\alpha }^{\circ r}\right]}_{jk}\right):=\left(o_k^{r\alpha }\right)\right\},\end{array}$$

(30)

which corresponds to the Abelian group generated by the operator $A_{\alpha }\left(o_x^{\alpha }\right)$ isomorphic to the group $\left(\mathbb{Z},+\right)$, that is,

$$\begin{array}{c}\hfill {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\cong \left(\mathbb{Z},+\right).\end{array}$$

(31)

Proof. 

It should be noted that due to the way the set (30) is defined, just the Hadamard product of type vertical is applied among its elements. So, $\forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ it is fulfilled that

$$\begin{array}{c}\hfill A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}=\left({\left[A_{\alpha }^{\circ p}\right]}_{jk}\right)\circ \left({\left[A_{\alpha }^{\circ q}\right]}_{jk}\right)=\left(o_k^{(p+q)\alpha }\right)=\left({\left[A_{\alpha }^{\circ (p+q)}\right]}_{jk}\right)=A_{\alpha }^{\circ (p+q)}.\end{array}$$

(32)

So, from the previous result, it is feasible to prove that the set ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ is a semigroup since it fulfills the following property:

$$\begin{array}{c}\hfill \forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q},A_{\alpha }^{\circ r}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\mathrm{it}\mathrm{is}\mathrm{fulfilled}\mathrm{that}\left(A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}\right)\circ A_{\alpha }^{\circ r}=A_{\alpha }^{\circ p}\circ \left(A_{\alpha }^{\circ q}\circ A_{\alpha }^{\circ r}\right).\end{array}$$

(33)

Furthermore, it follows from the definition of the set (30) that it contains a neutral element, with which it is feasible to prove from the previous result that the set ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ is also a monoid since it fulfills the following property:

$$\begin{array}{c}\hfill \exists A_{\alpha }^{\circ 0}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\mathrm{such}\mathrm{that}\forall A_{\alpha }^{\circ p}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\mathrm{it}\mathrm{is}\mathrm{fulfilled}\mathrm{that}{A}_{\alpha }^{\circ 0}\circ A_{\alpha }^{\circ p}=A_{\alpha }^{\circ p}.\end{array}$$

(34)

It should be noted that due to the way in which the set (30) is defined, for each element contained in the set its symmetric element is also defined, with which from the previous result the set ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ is also a group since it fulfills the following property:

$$\begin{array}{c}\hfill \forall A_{\alpha }^{\circ p}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\exists A_{\alpha }^{\circ -p}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\mathrm{such}\mathrm{that}{A}_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ -p}=A_{\alpha }^{\circ 0}.\end{array}$$

(35)

Finally, observing that the order in which the elements of the sets are operated does not influence the final result, it is obtained that the set ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ is also an Abelian group since it fulfills the following property:

$$\begin{array}{c}\hfill \forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\mathrm{it}\mathrm{is}\mathrm{fulfilled}\mathrm{that}{A}_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}=A_{\alpha }^{\circ q}\circ A_{\alpha }^{\circ p}.\end{array}$$

(36)

Once proven that the set (30) defines an Abelian group, to finish the proof of the theorem it is enough to define a bijective homomorphism between the sets ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ and $\left(\mathbb{Z},+\right)$. So, defining the following functions

$$\begin{array}{c}\hfill \begin{array}{ccc}\begin{array}{c}\psi :{}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\to \left(\mathbb{Z},+\right)\\ \psi \left(A_{\alpha }^{\circ r}\right)=r\end{array}\hfill & \mathrm{and}& \hfill \begin{array}{c}{\psi }^{-1}:\left(\mathbb{Z},+\right)\to {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\\ {\psi }^{-1}\left(r\right)=A_{\alpha }^{\circ r}\end{array}\end{array},\end{array}$$

(37)

it is feasible to prove that the function ψ defines a homeomorphism between the sets ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ and $\left(\mathbb{Z},+\right)$ through the following result:

$$\begin{array}{c}\hfill \forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q}\in {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)\mathrm{it}\mathrm{is}\mathrm{fulfilled}\mathrm{that}\psi \left(A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}\right)=\psi \left(A_{\alpha }^{\circ (p+q)}\right)=p+q=\psi \left(A_{\alpha }^{\circ p}\right)+\psi \left(A_{\alpha }^{\circ q}\right),\end{array}$$

(38)

and analogously it is proved that the function ${\psi }^{-1}$ defines a homeomorphism between the sets $\left(\mathbb{Z},+\right)$ and ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ through the following result:

$$\begin{array}{c}\hfill \forall p,q\in \left(\mathbb{Z},+\right)\mathrm{it}\mathrm{is}\mathrm{fulfilled}\mathrm{that}{\psi }^{-1}\left(p+q\right)=A_{\alpha }^{\circ (p+q)}=A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}={\psi }^{-1}\left(p\right)\circ {\psi }^{-1}\left(q\right).\end{array}$$

(39)

Therefore, from the previous results, it follows that the function ψ defines an isomorphism between the sets ${}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right)$ and $\left(\mathbb{Z},+\right)$. □

Therefore, from the previous theorem, it is feasible to obtain the following corollary:

Corollary 1.

Let $o_x^{\alpha }$ be a fractional operator such that $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$ and let $\left(\mathbb{Z},+\right)$ be the group of integers under the addition. Therefore, considering the modified Hadamard product given by (29) and some subgroup $\mathbb{H}$ of the group $\left(\mathbb{Z},+\right)$, it is feasible to define the following set of fractional matrix operators

$$\begin{array}{c}\hfill {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{H}\right):=\left\{A_{\alpha }^{\circ r}=A_{\alpha }\left(o_x^{r\alpha }\right):r\in \mathbb{H}and{A}_{\alpha }^{\circ r}=\left({\left[A_{\alpha }^{\circ r}\right]}_{jk}\right):=\left(o_k^{r\alpha }\right)\right\},\end{array}$$

(40)

which corresponds to a subgroup of the group generated by the operator $A_{\alpha }\left(o_x^{\alpha }\right)$, that is,

$$\begin{array}{c}\hfill {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{H}\right)\le {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right).\end{array}$$

(41)

Example 1.

Let ${\mathbb{Z}}_n$ be the set of residual classes less than a positive integer n. Therefore, considering a fractional operator $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$ and the set ${\mathbb{Z}}_{14}$, it is feasible to define, under the modified Hadamard product given by (29), the following Abelian group of fractional matrix operators

$$\begin{array}{c}\hfill {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{Z}}_{14}\right)=\left\{A_{\alpha }^{\circ 0},A_{\alpha }^{\circ 1},A_{\alpha }^{\circ 2},A_{\alpha }^{\circ 3},A_{\alpha }^{\circ 4},A_{\alpha }^{\circ 5},A_{\alpha }^{\circ 6},A_{\alpha }^{\circ 7},A_{\alpha }^{\circ 8},A_{\alpha }^{\circ 9},A_{\alpha }^{\circ 10},A_{\alpha }^{\circ 11},A_{\alpha }^{\circ 12},A_{\alpha }^{\circ 13}\right\}.\end{array}$$

(42)

Furthermore, all possible combinations of the elements of the group are summarized in the following Cayley table:

It is worth mentioning that the Corollary 1 allows generating groups of fractional operators under other operations. For example, considering the following operation

$$\begin{array}{c}\hfill A_{\alpha }^{\circ r}\ast A_{\alpha }^{\circ s}=A_{\alpha }^{\circ rs},\end{array}$$

(43)

it is feasible to obtain the following corollaries:

Corollary 2.

Let ${\mathbb{M}}_n$ be the set of positive residual classes corresponding to the coprimes less than a positive integer n. Therefore, for each fractional operator $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$, it is feasible to define the following Abelian group of fractional matrix operators under the operation (43):

$$\begin{array}{c}\hfill {}_m{G}^{*}\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{M}}_n\right):=\left\{A_{\alpha }^{\circ r}=A_{\alpha }\left(o_x^{r\alpha }\right):r\in {\mathbb{M}}_{n}and{A}_{\alpha }^{\circ r}=\left({\left[A_{\alpha }^{\circ r}\right]}_{jk}\right):=\left(o_k^{r\alpha }\right)\right\}.\end{array}$$

(44)

Example 2.

Let $o_x^{\alpha }$ be a fractional operator such that $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$. Therefore, considering the set ${\mathbb{M}}_{14}$, it is feasible to define, under the operation (43), the following Abelian group of fractional matrix operators

$$\begin{array}{c}\hfill {}_m{G}^{*}\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{M}}_{14}\right)=\left\{A_{\alpha }^{\circ 1},A_{\alpha }^{\circ 3},A_{\alpha }^{\circ 5},A_{\alpha }^{\circ 9},A_{\alpha }^{\circ 11},A_{\alpha }^{\circ 13}\right\}.\end{array}$$

(45)

Furthermore, all possible combinations of the elements of the group are summarized in the following Cayley table:

Corollary 3.

Let ${\mathbb{Z}}_p^{+}$ be the set of positive residual classes less than p, with p a prime number. Therefore, for each fractional operator $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$, it is feasible to define the following Abelian group of fractional matrix operators under the operation (43):

$$\begin{array}{c}\hfill {}_m{G}^{*}\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{Z}}_p^{+}\right):=\left\{A_{\alpha }^{\circ r}=A_{\alpha }\left(o_x^{r\alpha }\right):r\in {\mathbb{Z}}_p^{+}and{A}_{\alpha }^{\circ r}=\left({\left[A_{\alpha }^{\circ r}\right]}_{jk}\right):=\left(o_k^{r\alpha }\right)\right\}.\end{array}$$

(46)

Example 3.

Let $o_x^{\alpha }$ be a fractional operator such that $o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)$. Therefore, considering the set ${\mathbb{Z}}_{13}^{+}$, it is feasible to define, under the operation (43), the following Abelian group of fractional matrix operators

$$\begin{array}{c}\hfill {}_m{G}^{*}\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{Z}}_{13}^{+}\right)=\left\{A_{\alpha }^{\circ 1},A_{\alpha }^{\circ 2},A_{\alpha }^{\circ 3},A_{\alpha }^{\circ 4},A_{\alpha }^{\circ 5},A_{\alpha }^{\circ 6},A_{\alpha }^{\circ 7},A_{\alpha }^{\circ 8},A_{\alpha }^{\circ 9},A_{\alpha }^{\circ 10},A_{\alpha }^{\circ 11},A_{\alpha }^{\circ 12}\right\}.\end{array}$$

(47)

Furthermore, all possible combinations of the elements of the group are summarized in the following Cayley table:

Finally, it should be noted that when n is a prime number, the following result is obtained:

$$\begin{array}{c}\hfill {}_m{G}^{*}\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{M}}_n\right)={}_m{G}^{*}\left(A_{\alpha }\left(o_x^{\alpha }\right),{\mathbb{Z}}_n^{+}\right).\end{array}$$

(48)

4. Conclusions

Although this article presents one way to define groups of fractional operators using sets related to the set of integer numbers, it would be feasible to extend the results using other sets of numbers that allow defining Abelian groups, as is the case of the set of rational numbers and the set of real numbers, being feasible to define the following groups:

$$\begin{array}{ccc}\hfill {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{Q}\right)& \mathrm{and}& {}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{R}\right).\hfill \end{array}$$

(49)

Furthermore, from the groups generated by the equation (30), it is feasible define the following group of fractional matrix operators [42,44]:

$$\begin{array}{c}\hfill {}_m{G}_{FIM}\left(\alpha \right):=\bigcup _{o_x^{\alpha }\in {}_m{MO}_{x,\alpha }^{\infty ,u}\left(h\right)}{}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right)\right),\end{array}$$

(50)

in which it is assumed that through combinations of the horizontal and vertical type of the modified Hadamard product given by the equation (29), the fractional operators are reduced to their minimum expression, allowing to obtain $\forall A_{i,\alpha }^{\circ p},A_{j,\alpha }^{\circ q},A_{j,\alpha }^{\circ r}\in {}_m{G}_{FIM}\left(\alpha \right)$, with $i\ne j$, the following result:

$$\begin{array}{cc}\hfill \left(A_{i,\alpha }^{\circ p}\circ A_{j,\alpha }^{\circ q}\right)\circ A_{j,\alpha }^{\circ r}=A_{i,\alpha }^{\circ p}\circ \left(A_{j,\alpha }^{\circ q}\circ A_{j,\alpha }^{\circ r}\right)=A_{k,\alpha }^{\circ 1}:=A_{k,\alpha }\left(o_{i,x}^{p\alpha }\circ o_{j,x}^{(q+r)\alpha }\right),& p,q,r\in \mathbb{Z}\setminus \left\{0\right\}.\end{array}$$

(51)

As a consequence, the following result is obtained:

$$\begin{array}{c}\hfill \forall A_{k,\alpha }^{\circ 1}\in {}_m{G}_{FIM}\left(\alpha \right)\mathrm{such}\mathrm{that}{A}_{k,\alpha }\left(o_{k,x}^{\alpha }\right)=A_{k,\alpha }\left(o_{i,x}^{p\alpha }\circ o_{j,x}^{q\alpha }\right)\exists A_{k,\alpha }^{\circ r}=A_{k,\alpha }^{\circ (r-1)}\circ A_{k,\alpha }^{\circ 1}=A_{k,\alpha }\left(o_{i,x}^{rp\alpha }\circ o_{j,x}^{rq\alpha }\right).\end{array}$$

(52)

Therefore, if ${\Phi }_{FIM}$ denotes the iteration function of some fractional iterative method [43,44], it is feasible to obtain the following result:

$$\begin{array}{c}\hfill \mathrm{Let}{\alpha }_0\in \mathbb{R}\setminus \mathbb{Z}\Rightarrow \forall A_{{\alpha }_0}^{\circ 1}\in {}_m{G}_{FIM}\left(\alpha \right)\exists {\Phi }_{FIM}={\Phi }_{FIM}\left(A_{{\alpha }_0}\right)\therefore \forall A_{{\alpha }_0}\exists \left\{{\Phi }_{FIM}\left(A_{\alpha }\right):\alpha \in \mathbb{R}\setminus \mathbb{Z}\right\}.\end{array}$$

(53)

Finally, it is worth mentioning that it is feasible to develop more complex algebraic structures of fractional operators using the presented results. For example, without loss of generality, considering the modified Hadamard product (29) and the operation (43), a commutative and unitary ring of fractional operators may be defined as follows

$$\begin{array}{c}\hfill {}_{m}R\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{R}\right):=\left({}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{R}\right),\circ ,\ast \right),\end{array}$$

(54)

in which it is not difficult to verify the following properties:

• The pair $\left({}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{R}\right),\circ \right)$ is an Abelian group.
• The pair $\left({}_{m}G\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{R}\right),\ast \right)$ is a commutative monoid.
• $\forall A_{\alpha }^{\circ p},A_{\alpha }^{\circ q},A_{\alpha }^{\circ r}\in {}_{m}R\left(A_{\alpha }\left(o_x^{\alpha }\right),\mathbb{R}\right)$, the operation ∗ is distributive with respect to the operation ∘, that is,

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_{\alpha }^{\circ p}\ast \left(A_{\alpha }^{\circ q}\circ A_{\alpha }^{\circ r}\right)=\left(A_{\alpha }^{\circ p}\ast A_{\alpha }^{\circ q}\right)\circ \left(A_{\alpha }^{\circ p}\ast A_{\alpha }^{\circ r}\right)\\ \left(A_{\alpha }^{\circ p}\circ A_{\alpha }^{\circ q}\right)\ast A_{\alpha }^{\circ r}=\left(A_{\alpha }^{\circ p}\ast A_{\alpha }^{\circ r}\right)\circ \left(A_{\alpha }^{\circ q}\ast A_{\alpha }^{\circ r}\right)\end{array}\right..\end{array}$$

  (55)