Multi-Kernel General Fractional Calculus of Arbitrary Order

Abstract

An extension of the general fractional calculus (GFC) of an arbitrary order, proposed by Luchko, is formulated. This extension is also based on a multi-kernel approach, in which the Laplace convolutions of different Sonin kernels are used. The proposed multi-kernel GFC of an arbitrary order is also considered for the case of intervals $(a,b)$ where $-\infty <a<b\le \infty$. Examples of multi-kernel general fractional operators of arbitrary orders are proposed.

1. Introduction

Fractional calculus is a mathematical theory of integral and integro-differential operators, which are interpreted as integrals and derivatives of arbitrary orders [1,2,3,4,5,6,7]. This interpretation is based on the fulfillment of generalized analogs of the first and second fundamental theorems of the standard mathematical calculus, which pertain to derivatives and integrals of the integer order. Fractional calculus is widely used to describe various phenomena and processes with nonlocality in space and time (see books [8,9,10,11,12,13,14,15,16,17] and handbooks [18,19]). However, fractional calculus mainly uses operator kernels of the power-law type.

To describe a wider range of nonlocalities, it is important to expand the class of integral and integro-differential operators, for which analogs of the first and second fundamental theorems of the standard mathematical calculus can be realized. One of the most interesting and urgent directions is the so-called general fractional calculus, which is based on the development and generalization of Sonin’s approach [20,21]. The basic concepts of new mathematical calculus were proposed by Sonin in his 1884 paper [20] (see also [21]). The terms “general fractional calculus”, “general fractional integrals”, and “general fractional derivatives” were suggested by Kochubei in 2011 [22].

Following work [23], one can conditionally distinguish three types of general fractional calculus (GFC), which are based on the three classes of Sonin kernels: (K) Kochubei’s GFC, proposed in 2011 [22] (see also [24,25]); (H) Hanyga’s GFC, proposed in 2020 [26]; and (L) Luchko’s GFC, proposed in 2021 [23].

During the past two years, Luchko’s general fractional (GF) calculus has been developed very actively in works [23,27,28,29,30,31,32,33,34]. The n-fold GF integrals and GF derivatives of the first order were proposed in [23]. The general fractional calculus of an arbitrary order was first proposed in Luchko’s works [27,31] (see also [29,30]). A composition of GF integrals with identical kernels and n-fold sequential GF derivatives with identical kernels were defined in [29,30], where solutions to equations with these operators were derived. For GF differential and GF integral operators of arbitrary orders, the fundamental theorems of the general fractional calculus were proved. A convolution series in GFC [29,30] and a general fractional generalization of the Taylor series were proposed in [30]. Generalization of the operational calculus for equations with general fractional derivatives was suggested [28,33]. Note that an attempt to generalize Luchko’s GFC to the multi-kernel case was made in work [35]. The general fractional calculus with functions of a several variable (the multivariable) GFC and the general fractional vector calculus (GFVC) were realized in work [36]. Trends in the development and applications of GFC and some open problems of GFC are described in Section 2 and Section 4 of the review paper [37].

The applications and development of general fractional calculus can be divided conditionally into applications in mathematics, which will include both the development of the calculus itself and its applications in other sciences.

Among the considered applications and development of GFC in mathematics, one can note results in the following directions: the theory of integral equations of the first kind [38,39]; subordinators and their inverse processes (hitting times) that are presented by governing equations with convolution-type integro-differential operators [40]; uniqueness and the existence of solutions in the initial boundary value problems for a general time-fractional diffusion equation [41] and other types of fractional partial differential equations [42]; fractional ordinary differential equations with the GFD [42]; the well-posedness of general Caputo-type fractional differential equations [43]; abstract Cauchy problems for generalized fractional differential equations and a generalized Gronwall inequality [44]; the Prabhakar (three-parameter Mittag–Leffler) function in the general fractional calculus [45]; a general fractional relaxation equation and its application to an inverse source problem [46]; identification of the source term in a nonlocal problem for the general time-fractional diffusion equation [47]; and the nonlocal (general fractional) probability theory [48].

Among the considered applications of GFC in physics and other sciences, one can note the following areas: fractional statistical dynamics of continuously interacting particle systems [49]; the fractional relaxation process [42,49]; the fractional growth process [42,50]; general fractional dynamics as an interdisciplinary science, in which the nonlocal properties of dynamical systems are studied by using general fractional calculus and general nonlocal maps [51]; the asymptotic behavior of random time changes of general fractional dynamical systems and Cesaro limits [52]; inverse problems for generalized subdiffusion processes [53,54] thermodynamic restrictions for a general Zener model [55]; a two-compartment pharmacokinetic model [56]; general non-Markovian quantum dynamics of open and dissipative systems [57]; general nonlocal continuum mechanics with nonlocal balance laws [58], which are based on the general fractional vector calculus proposed in [36]; general nonlocal electrodynamics with general fractional Maxwell equations and nonlocal effects [59]; the nonlocal classical field theory of gravity with the effects of massiveness of the nonlocality and mass shielding by nonlocality [60]; and nonlocal statistical mechanics with general fractional Liouville equations [61], which are based on the nonlocal probability theory proposed in [48].

The following are the mathematical motivations for this article: extensions of the GFC of an arbitrary order on a multi-kernel case, the finite intervals of the real line as generalizations of Luchko’s m-fold GFC of an arbitrary order [27], and a multi-kernel approach to GFC [35]. Let us describe these mathematical motivations in some detail.

Let us first describe in more detail the motivation for the multi-kernel expansion of general fractional calculus (GFC). Luchko’s form of the GFC, described in works [23,27,28,29,30,31,32,33,34], can be called a single-kernel GFC. Let us briefly list those general fractional operators whose definitions are given and whose properties are described in these works by Luchko. The general fractional integrals (GFIs) and general fractional derivatives (GFDs) of arbitrary orders are defined by Luchko in paper [27]. The main basic properties of GFIs and GFDs of arbitrary orders are described by the two fundamental theorems of GFC, which are proved in [27]. The m-fold GFI and GFDs with the kernel pairs $(M,K)\in {\mathcal{L}}_1$ are proposed by Luchko in [29,30]. Then, these operators are generalized for the kernel pairs $(M,K)\in {\mathcal{L}}_n$ with $n\in \mathbb{N}$ in Luchko’s work [31]. The following GF operators are constructed as convolutional powers of the same kernel: (a) The m-fold GFIs of arbitrary orders can be represented as the GFI with the kernel $M^{<m>}\left(x\right)=(M_1\ast \dots \ast M_m)\left(x\right)$. (b) The m-fold GFDs of arbitrary orders can be represented as the GFI with the kernel $K^{<m>}\left(x\right)=(K_1\ast \dots \ast K_m)\left(x\right)$, where $M_j\left(x\right)=M\left(x\right)$ and $K_j\left(x\right)=K\left(x\right)$ for all $j=1,\dots ,m$, and $(M,K)\in {\mathcal{L}}_n$.

It is important to extend the general fractional calculus by using the convolutional product of different kernels instead of the convolutional degree. Therefore, one can consider kernels in the forms $(M_1\ast \dots \ast M_m)\left(x\right)$ and $(K_1\ast \dots \ast K_m)\left(x\right)$, where the conditions $M_j\left(x\right)=M\left(x\right)$ and $K_j\left(x\right)=K\left(x\right)$ for all $j=1,\dots ,m$ are not used. The first attempt to construct such multi-kernel fractional calculus was made in paper [35] in 2021. However, the proposed subsets of multi-kernel general fractional operators have drawbacks. For example, it should be noted that the operators, which are defined by Equations (53) and (55) in [35], cannot be considered fractional derivatives in the general case (for all values of parameters). For some values of the parameters, these operators are described by sequential actions of the general fractional derivative and integral of the integer order.

Therefore, it is important to remove these drawbacks to derive the correct formulation of the multi-kernel GFC. This can be realized by taking into account the results of Luchko’s works [27,31] and [29,30]. For the correct formulation of the multi-kernel approach to general fractional calculus, one can generalize Luchko’s results, which are derived in [31], for a set of kernel pairs $(M_j,K_j)\in {\mathcal{L}}_n$ with $n\in \mathbb{N}$, where $j=1,\dots ,m$.

Because of this, the motivation of this paper is the correct formulation of the multi-kernel GFC of an arbitrary order, including the definitions of the multi-kernel general fractional operators of arbitrary orders and the proof of the fundamental theorems of the multi-kernel GFC of an arbitrary order. Then, the single-kernel m-fold sequential GFC of an arbitrary order, which is proposed by Luchko for the case $M_j\left(x\right)=M\left(x\right)$ and $K_j\left(x\right)=K\left(x\right)$ for $j=1,\dots ,m$, can be considered a special case of the multi-kernel GFC.

Let us briefly describe the content of this article.

In Section 2 (preliminaries), Luchko’s results for the single-kernel GFC of an arbitrary order are described to simplify further consideration. In Section 3, the multi-kernel GFC of an arbitrary order is proposed for the interval $(0,\infty )$. In Section 4, the multi-kernel GF operators of arbitrary orders for the finite intervals $[a,b]$ are proposed. In Section 5, examples of the multi-kernel GF operators of arbitrary orders are considered. A brief conclusion is given in Section 6.

2. Preliminaries: Single-Kernel GFC of Arbitrary Order

In this section, some basic definitions and properties of the GFIs and GFDs are described as in [23].

2.1. Spaces and Sub-Spaces of Functions

Let us define the function spaces $C_{-1}(0,\infty )$ and $C_{-1,0}(0,\infty )$ [23,62].

Definition 1.

Let the function $f\left(x\right)$ be represented in the form

$$f\left(x\right)=x^{p}g\left(x\right),(p>-1)$$

(1)

for all $x>0$, where $g\left(x\right)\in C[0,\infty )$.

This set of functions is denoted as $C_{-1}(0,\infty )$.

Definition 2.

Let the function $K\left(x\right)$ be represented in the form

$$K\left(x\right)=x^{q}k\left(x\right),(-1<q<0)$$

(2)

for all $x>0$, where $k\left(x\right)\in C[0,\infty )$.

This set of functions is denoted as $C_{-1,0}(0,\infty )$.

Note that the inclusion $C_{-1,0}(0,\infty )\subseteq C_{-1}(0,\infty )$ holds true. Note also that the inclusion $C(0,\infty )\subseteq C_{-1}(0,\infty )$ holds true since $C(0,\infty )=C_0(0,\infty )$ [62].

One can also consider the spaces $C_{-1}^n(0,\infty )$ with $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$.

Definition 3.

Let the derivative $f^{\left(n\right)}\left(x\right)=d^{n}f\left(x\right)/d{x}^n$ of the integer order $n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ of the function $f\left(x\right)$ belong to the space $C_{-1}(0,\infty )$, i.e., $f^{\left(n\right)}\left(x\right)\in C_{-1}(0,\infty )$.

This set of functions is denoted as $C_{-1}^n(0,\infty )$, where $C_{-1}^0(0,\infty )\equiv C_{-1}(0,\infty )$.

The following theorem describes properties of the space $C_{-1}^n(0,\infty )$.

Theorem 1

(Theorem Luchko–Gorenflo). The set

$$C_{-1}^n(0,\infty ):=\{f\left(x\right):f^{\left(n\right)}\left(x\right)\in C_{-1}(0,\infty )\}$$

(3)

satisfies the following properties.

1. 

$C_{-1}^n(0,\infty )$ with $n\in {\mathbb{N}}_0$ is a vector space over the field $\mathbb{R}$.

2. 

If $F\left(x\right)\in C_{-1}^n(0,\infty )$ with $n\ge 1$, then

$$F^{\left(k\right)}(0+):=\underset{t\to 0+}{lim}{F}^{\left(k\right)}\left(x\right)<\infty ,(0\le k\le n-1,k\in {\mathbb{N}}_0).$$

(4)

3. 

If $F\left(x\right)\in C_{-1}^n(0,\infty )$ with $n\ge 1$, then

$$F^{\ast }\left(x\right)=\left\{\begin{array}{cc}F\left(x\right),\hfill & x>0,\hfill \\ F(0+),\hfill & x=0\hfill \end{array}\right.$$

(5)

belongs to the space $C^{n-1}[0,\infty )$.

4. 

If $F\left(x\right)\in C_{-1}^n(0,\infty )$ with $n\ge 1$, then $F\left(x\right)\in C^n(0,\infty )\cap C^{n-1}[0,\infty )$.

5. 

In order for $F\left(x\right)\in C_{-1}^n(0,\infty )$ with $n\ge 1$, it is necessary and sufficient that $F\left(x\right)$ can be represented in the form

$$F\left(x\right)=\left(I_{0+}^n\phi \right)\left(x\right)+\sum _{k=0}^{n-1}{F}^{\left(k\right)}\left(0\right)\frac{x^k}{k!},$$

(6)

for all $x\ge 0$, where $\phi \in C_{-1}(0,\infty )$ and where $I_{0+}^n$ is the standard integral of the integer order $n\in \mathbb{N}$ in the form

$$\left(I_{0+}^n\phi \right)\left(x\right)=\frac{1}{(n-1)!}{\int }_0^x{(x-u)}^{n-1}\phi \left(u\right)du.$$

(7)

Therefore, one can write

$$\left\{F\left(x\right)\in C_{-1}^n(0,\infty )n\ge 1\right\}\iff \left\{F\left(x\right)=\left(I_{0+}^n\phi \right)\left(x\right)+\sum _{k=0}^{n-1}{F}^{\left(k\right)}\left(0\right)\frac{x^k}{k!}\right\}.$$

(8)

6. 

Let $F\left(x\right)\in C_{-1}^n(0,\infty )$ with $n\in {\mathbb{N}}_0$, $K\left(x\right)\in C_{-1}^1(0,\infty )$, and

$$F^{\left(k\right)}\left(0\right)=0(k=0,\dots ,n-1).$$

(9)

Then, the Laplace convolution $z\left(x\right)=(K\ast F)\left(x\right)$ belongs to the space $C_{-1}^{n+1}(0,\infty )$, and $z^{\left(k\right)}\left(0\right)=0$ for $k=0,\dots ,n-1$.

Theorem 1 is proved in paper [62] (see also [23]).

2.2. General Fractional Integrals and Derivatives of Arbitrary Orders

The general fractional integrals (GFIs) and general fractional derivatives (GFDs) of arbitrary orders are defined by Luchko in his paper [27]. To define the GFIs and the GFDs of arbitrary orders, the following condition on their kernels $M\left(x\right)$ and $K\left(x\right)$ should be satisfied [27] in the form

$$(M\ast K)\left(x\right)={\left\{1\right\}}^n\left(x\right)\mathrm{for}\mathrm{all}x>0,$$

(10)

where $n\in \mathbb{N}$ and

$${\left\{1\right\}}^n\left(x\right):=(g_1\ast \dots \ast g_n)\left(x\right)=h_n\left(x\right)=\frac{x^{n-1}}{(n-1)!},$$

where $g_j\left(x\right)=h\left(x\right)=\left\{1\right\}$ are the Heaviside step functions for all $j=1,\dots ,n$. Condition (10), proposed in [27], is called the Luchko condition. Note that the Luchko condition (10) generalizes the Sonin condition $(M\ast K)\left(x\right)=\left\{1\right\}$, which holds for all $x>0$ where $h\left(x\right)=\left\{1\right\}$ is the Heaviside step function. The Sonin condition corresponds to the case $n=1$.

Let us define a set of the kernels that satisfies condition (10) and belongs to the special spaces of functions.

Definition 4.

Let the functions $M\left(x\right)$ and $K\left(x\right)$ satisfy the following conditions.

(1) $M\left(x\right)$ and $K\left(x\right)$ satisfy the Luchko condition (10).

(2) The kernel $M\left(x\right)$ of the GFI belongs to the space $C_{-1}(0,\infty )$.

(3) The kernel $K\left(x\right)$ of the GFD belongs to the space $C_{-1,0}(0,\infty )$.

A set of the pair $(M,K)$ of such kernels is called the Luchko set, which is denoted by ${\mathcal{L}}_n$.

Example 1.

The pair of kernels

$$M\left(x\right)=h_{\alpha }\left(x\right)=\frac{x^{\alpha -1}}{\Gamma \left(\alpha \right)},K\left(x\right)=h_{n-\alpha }\left(x\right)=\frac{x^{n-\alpha -1}}{\Gamma (n-\alpha )}$$

with $\alpha \in (n-1,n)$ and $n\in \mathbb{N}$ belongs to the Luchko set ${\mathcal{L}}_n$.

Let us consider the pair $(M_n,K_n)$ of kernels from the Luchko set ${\mathcal{L}}_n$ with $n>1$, which are built from the Sonin kernels $(M,K)$ from the Luchko set ${\mathcal{L}}_1$.

Theorem 2.

Let $(M,K)$ be a pair of Sonin kernels from the Luchko set ${\mathcal{L}}_1$.

Then, the pair $(M_n,K_n)$ of kernels, given by the equations

$$M_n\left(x\right)=({\left\{1\right\}}^{n-1}\ast M)\left(x\right),K_n\left(x\right)=K\left(x\right),$$

(11)

belongs to the Luchko set ${\mathcal{L}}_n$.

Proof. 

Theorem 2 is proved in [27]. □

Note that the transformations proving the fulfillment of the Luchko condition (10) for kernels (11) have the form

$$(M_n\ast K_n)\left(x\right)=({\left\{1\right\}}^{n-1}\ast M\ast K)\left(x\right)=({\left\{1\right\}}^{n-1}\ast \left\{1\right\})\left(x\right)={\left\{1\right\}}^n\left(x\right).$$

(12)

Let us define the GF integrals and GF derivatives of an arbitrary order and describe their basic properties.

Definition 5.

Let $(M,K)$ be a pair of kernels from the Luchko set ${\mathcal{L}}_n$, and let the function $f\left(x\right)$ belong to the set $C_{-1}(0,\infty )$.

The GFI with the kernel $M\left(x\right)$ is defined by the equation

$$I_{\left(M\right)}^x\left[u\right]f\left(u\right):={\int }_0^{x}M(x-u)f\left(u\right)du,$$

(13)

where $x>0$.

Definition 6.

Let $(M,K)$ be a pair of kernels from the Luchko set ${\mathcal{L}}_n$.

The GFD of the Riemann–Liouville type with the kernel $K\left(x\right)$ for the function $f\left(x\right)\in C_{-1}(0,\infty )$ is defined as follows:

$$D_{\left(K\right)}^x\left[u\right]f\left(u\right):=\frac{d^n}{d{x}^n}{\int }_0^{x}K(x-u)f\left(u\right)du,$$

(14)

where $x>0$.

The GFD of the Caputo type with the kernel $K\left(x\right)$ for the function $f^{\left(n\right)}\left(x\right)\in C_{-1}(0,\infty )$ is defined as

$$D_{\left(K\right)}^{x,\ast }\left[u\right]f\left(u\right):=I_{\left(K\right)}^x\left[u\right]f^{\left(n\right)}\left(u\right)={\int }_0^{x}K(x-u)f^{\left(n\right)}\left(u\right)du,$$

(15)

where $x>0$.

In [27], the GFD of the Caputo type (15) is defined by the equation

$$D_{\left(K\right)}^{x,\ast }\left[u\right]f\left(u\right):=D_{\left(K\right)}^x\left[u\right]\left(f\left(u\right)-\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(0\right)h_{k+1}\left(u\right)\right),$$

(16)

where $x>0$. Using Equation (16), the GFD of the Caputo type (16) can be represented [27] in the form

$$D_{\left(K\right)}^{x,\ast }\left[u\right]f\left(u\right)=D_{\left(K\right)}^x\left[u\right]f\left(u\right)-\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(0\right)K^{(n-k-1)}\left(x\right),$$

(17)

where $x>0$ and $K^{\left(m\right)}\left(x\right)=d^{m}K\left(x\right)/d{x}^m$.

Example 2.

The GFI (13) with the kernel $M\left(x\right)=h_{\alpha }\left(x\right)$, $\alpha >0$ is the Riemann–Liouville fractional integral [4].

The GFDs (14) and (15) with the kernel $K\left(x\right)=h_{n-\alpha }\left(x\right)$ are the Riemann–Liouville and Caputo fractional derivatives of the order $\alpha \in (n-1,n)$, where $n\in \mathbb{N}$ [4].

Let us give some important properties of the GFIs of arbitrary orders.

• If the function $f\left(x\right)$ belongs to the set $C_{-1}(0,\infty )$, then the GFI of this function also belongs to the set, i.e., $I_{\left(M\right)}^x\left[u\right]f\left(u\right)\in C_{-1}(0,\infty )$,

  $$I_{\left(M\right)}^x:C_{-1}(0,\infty )\to C_{-1}(0,\infty ),$$

  (18)
• For $f\left(x\right)\in C_{-1}(0,\infty )$, the following semi-group property is satisfactory

  $$I_{\left(M_1\right)}^x\left[u\right]I_{\left(M_2\right)}^u\left[w\right]f\left(w\right):=I_{(M_1\ast M_2)}^x\left[u\right]f\left(u\right).$$

  (19)

Remark 1.

Let $(M_1,K_1)$ be a pair of kernels from the Luchko set ${\mathcal{L}}_1$. Then, the GFI with the kernel $M_n=({\left\{1\right\}}^{n-1}\ast M_1)\left(x\right)$ can be written in the form

$$\left(I_{\left(M_n\right)}^{x}f\right)\left(x\right)=I_{0+}^{n-1}{I}_{\left(M_1\right)}^x\left[u\right]f\left(u\right)$$

(20)

for $x>0$. The GFDs of the Riemann–Liouville and Caputo types with the kernel $K_n\left(x\right)=K_1\left(x\right)$ can be represented as

$$D_{\left(K_n\right)}^x\left[u\right]f\left(u\right)=\frac{d^n}{d{x}^n}{I}_{\left(K_1\right)}^x\left[u\right]f\left(u\right),$$

(21)

$$D_{\left(K_n\right)}^{x,\ast }\left[u\right]f\left(u\right)=I_{\left(K_1\right)}^x\left[u\right]f^{\left(n\right)}\left(u\right)=\frac{d^n}{d{x}^n}{I}_{\left(K_1\right)}^x\left[u\right]\left(f\left(u\right)-\sum _{k=0}^{n-1}{f}^{\left(k\right)}\left(0\right)h_{k+1}\left(u\right)\right)$$

(22)

for $x>0$.

2.3. Fundamental Theorems of General Fractional Calculus of Arbitrary Order

The main basic properties of GFIs and GFDs of arbitrary orders are described by the two fundamental theorems of GFC, which are proved by Luchko in his paper [27].

The first fundamental theorem of GFC is formulated for the space $C_{-1,\left(K\right)}(0,\infty )$, which is defined by the following definition.

Definition 7.

Let the function $K\left(x\right)$ belong to the space $C_{-1}(0,\infty )$, and let the function $f\left(x\right)$ be represented in the form

$$f\left(x\right)=I_{\left(K\right)}^x\left[u\right]\phi \left(u\right),$$

(23)

for all $x>0$, where $\phi \left(x\right)\in C_{-1}(0,\infty )$.

Then, this set of functions, $f\left(x\right)$, is denoted as $C_{-1,\left(K\right)}(0,\infty )$.

Note that if $K\left(x\right)\in C_{-1,0}(0,\infty )$, then $K\left(x\right)\in C_{-1}(0,\infty )$ since $C_{-1,0}(0,\infty )\subset C_{-1}(0,\infty )$.

If $K\left(x\right)\in C_{-1}(0,\infty )$ and $\phi \left(x\right)\in C_{-1}(0,\infty )$, then $f\left(x\right)=I_{\left(K\right)}^x\left[u\right]\phi \left(u\right)\in C_{-1}(0,\infty )$.

Theorem 3

(First Fundamental Theorem of GFD of Arbitrary Order). Let $(M,K)$ be a pair of kernels from the Luchko set ${\mathcal{L}}_n$.

The GFD of the Riemann–Liouville type is a left inverse operator to the GFI, and the equation

$$D_{\left(K\right)}^x\left[u\right]I_{\left(M\right)}^u\left[w\right]f\left(w\right)=f\left(x\right)$$

(24)

holds for all $x>0$, if the function $f\left(x\right)$ belongs to the space $C_{-1}(0,\infty )$.

The GFD of the Caputo type is a left inverse operator to the GFI, and the equation

$$D_{\left(K\right)}^{x,\ast }\left[u\right]I_{\left(M\right)}^u\left[w\right]f\left(w\right)=f\left(x\right),$$

(25)

holds for all $x>0$, if the function $f\left(x\right)$ belongs to the space $C_{-1,\left(K\right)}(0,\infty )$.

Proof. 

Theorem 3 is proved in the same way as Theorem 4 in [27], pp. 11–12.

For the GFD of the Riemann–Liouville type, the following transformations can be used

$$D_{\left(K\right)}^x\left[u\right]I_{\left(M\right)}^u\left[w\right]f\left(u\right)=\frac{d^n}{d{x}^n}(K\ast (M\ast f))\left(x\right)=\frac{d^n}{d{x}^n}((K\ast M)\ast f)\left(x\right)=$$

$$\frac{d^n}{d{x}^n}({\left\{1\right\}}^n\ast f)\left(x\right)=\frac{d^n}{d{x}^n}\left(I_{0+}^{n}f\right)\left(x\right)=f\left(x\right),$$

where $I_{0+}^n$ is the standard integral of the integer order $n\in \mathbb{N}$ that has the form (7).

For the GFD of the Caputo type, the following properties and transformations are used. A function $f\left(x\right)\in C_{-1,\left(K\right)}(0,\infty )$ can be represented in the form $f\left(x\right)=I_{\left(K\right)}^x\left[u\right]\phi \left(u\right)$, where $\phi \left(x\right)\in C_{-1}(0,\infty )$. Therefore, the following transformations are valid:

$$I_{\left(M\right)}^x\left[u\right]f\left(u\right)=I_{\left(M\right)}^x\left[u\right]I_{\left(K\right)}^u\left[w\right]\phi \left(w\right)=((M\ast K)\ast \phi )\left(x\right)=({\left\{1\right\}}^n\ast \phi )\left(x\right)=\left(I_{0+}^n\phi \right)\left(x\right).$$

As a result, one can see that $I_{\left(M\right)}^x\left[u\right]f\left(u\right)\in C_{-1}^n(0,\infty )$, and the property

$$(\frac{d^k}{d{x}^k}{I}_{\left(M\right)}^x\left[u\right]f\left(u\right){)}_{x=0}={\left(\left(I_{0+}^{n-k}\phi \right)\left(x\right)\right)}_{x=0}=0$$

(26)

is satisfied for all $k=0,\dots ,n-1$. Equation (26) is given as Equation (63) in [27], p. 11.

Here, we use the property according to which, for a non-negative continuous function $\phi \left(x\right)$ on an open interval $(0,b)$ with $b>0$, the following limit exists and is equal to zero

$$\underset{x\to 0+}{lim}{\int }_0^x{g}_k\left(u\right)du=0,$$

(27)

where $g_k\left(u\right)=I_{0+}^{n-k-1}\left[w\right]\phi \left(w\right)$ and $k=0,\dots ,n-1$.

Using (17) of the GFD of the Caputo type, Equations (26) and (24) give

$$D_{\left(K\right)}^{x,\ast }\left[u\right]I_{\left(M\right)}^u\left[w\right]f\left(w\right)=D_{\left(K\right)}^x\left[u\right]I_{\left(M\right)}^u\left[w\right]f\left(w\right)-\sum _{k=0}^{n-1}{\left(\frac{d^k}{d{x}^k}{I}_{\left(M\right)}^x\left[u\right]f\left(u\right)\right)}_{x=0}\frac{d^{n-k-1}}{d{x}^{n-k-1}}K\left(x\right)=f\left(x\right)$$

for all $x>0$. □

The second fundamental theorem of the GFC can be formulated for the spaces $C_{-1}^n(0,\infty )$ and $C_{-1,\left(M\right)}(0,\infty )$ for the GFDs of the Caputo and Riemann–Liouville types, respectively.

Theorem 4

(Second Fundamental Theorem of Caputo-Type GFD of Arbitrary Order). Let $(M,K)$ be a pair of kernels from the Luchko set ${\mathcal{L}}_n$, and let the function $F\left(x\right)$ belong to the space $C_{-1}^n(0,\infty )$, i.e., $F^{\left(n\right)}\left(x\right)\in C_{-1}(0,\infty )$ for all $x>0$.

Then, for the GFD of the Caputo type, the equation

$$I_{\left(M\right)}^x\left[u\right]D_{\left(K\right)}^{u,\ast }\left[w\right]F\left(w\right)=F\left(x\right)-\sum _{k=0}^{n-1}{F}^{\left(k\right)}\left(0\right)h_{k+1}\left(x\right)$$

(28)

holds for all $x>0$, where $F^{\left(k\right)}\left(x\right)=d^{k}F\left(x\right)/d{x}^k$.

Proof. 

Theorem 4 is proved in the same manner as Theorem 5 in [27], p. 12.

Using the premise that any function $F\left(x\right)$ from the space $C_{-1}^n(0,\infty )$ can be represented (see Equation (6) of Theorem 1 and [62]) in the form

$$F\left(x\right)=\left(I_{0+}^n\phi \right)\left(x\right)+\sum _{k=0}^{n-1}{F}^{\left(k\right)}\left(0\right)h_{k+1}\left(x\right)$$

(29)

for all $x\ge 0$, where $\phi \left(x\right)\in C_{-1}(0,\infty )$.

Then, using Equation (16) and the definition of the GFD of the Caputo type in the form (29), one can realize the following transformations

$$D_{\left(K\right)}^{x,\ast }\left[u\right]F\left(u\right)=D_{\left(K\right)}^x\left[u\right]\left(F\left(u\right)-\sum _{k=0}^{n-1}{F}^{\left(k\right)}\left(0\right)h_{k+1}\left(u\right)\right)=D_{\left(K\right)}^x\left[u\right]I_{0+}^n\phi \left(u\right)=$$

$$\frac{d^n}{d{x}^n}(K\ast {\left\{1\right\}}^n\ast \phi )\left(x\right)=\frac{d^n}{d{x}^n}({\left\{1\right\}}^n\ast (K\ast \phi ))\left(x\right)=(K\ast \phi )\left(x\right).$$

Using representation (29), one can implement transformations

$$I_{\left(M\right)}^x\left[u\right]D_{\left(K\right)}^{u,\ast }\left[w\right]F\left(w\right)=I_{\left(M\right)}^x\left[u\right](K\ast \phi )\left(u\right)=((M\ast K)\ast \phi )\left(x\right)=$$

$$({\left\{1\right\}}^n\ast \phi )\left(x\right)=\left(I_{0+}^n\phi \right)\left(x\right)=F\left(x\right)-\sum _{k=0}^{n-1}{F}^{\left(k\right)}\left(0\right)h_{k+1}\left(x\right).$$

This proves equality (28). □

Theorem 5

(Second Fundamental Theorem of RL-Type GFD of Arbitrary Order). Let $(M,K)$ be a pair of kernels from the Luchko set ${\mathcal{L}}_n$, and let the function $F\left(x\right)$ belong to the set $C_{-1,\left(M\right)}(0,\infty )$.

Then, for the GFD of the Riemann–Liouville type, the equation

$$I_{\left(M\right)}^x\left[u\right]D_{\left(K\right)}^u\left[w\right]F\left(u\right)=F\left(x\right)$$

(30)

holds for all $x>0$.

Proof. 

Theorem 4 is proved in [27], p. 12.

The function $F\left(x\right)\in C_{-1,\left(M\right)}(0,\infty )$ can be represented in the form $F\left(x\right)=I_{\left(M\right)}^x\left[u\right]f\left(u\right)$, where $f\left(x\right)\in C_{-1}(0,\infty )$. Then, the following transformations can be realized

$$I_{\left(M\right)}^x\left[u\right]D_{\left(K\right)}^u\left[w\right]F\left(w\right)=I_{\left(M\right)}^x\left[u\right]\frac{d^n}{d{u}^n}(K\ast F)\left(u\right)=I_{\left(M\right)}^x\left[u\right]\frac{d^n}{d{u}^n}(K\ast (M\ast f))\left(u\right)=$$

$$I_{\left(M\right)}^x\left[u\right]\frac{d^n}{d{u}^n}({\left\{1\right\}}^n\ast f)\left(u\right)=I_{\left(M\right)}^x\left[u\right]f\left(u\right)=F\left(x\right).$$

This proves equality (30). □

2.4. Sequential n-Fold General Fractional Integrals and Derivatives

Let us consider sequential general fractional integrals and derivatives. The m-fold GFI and GFDs with the kernels $(M,K)\in {\mathcal{L}}_1$ are proposed by Luchko in [29,30]. Then, these operators are generalized for the kernels $(M,K)\in {\mathcal{L}}_n$ with $n\in \mathbb{N}$ in Luchko’s work [31].

Let us define the convolution power $f^{<m>}\left(x\right)$ with $m\in \mathbb{N}$ of the function $f\left(x\right)\in C_{-1}(0,\infty )$ by the equation

$$f^{<m>}\left(x\right):=(f_1\ast \dots \ast f_m)\left(x\right),$$

(31)

where $f_j\left(x\right)=f\left(x\right)$ for all $j=1,\dots ,m$.

Let us define the m-fold sequential GFI and GFD of the Riemann–Liouville type.

Definition 8.

Let the kernel pairs $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$.

The m-fold sequential GFI is defined as a composition of m GFIs with the kernel $M\left(x\right)$ in the form

$$I_{\left(M\right)}^{<m>,x}\left[u\right]f\left(u\right):=I_{\left(M\right)}^x\left[x_1\right]\dots I_{\left(M\right)}^{x_{m-1}}\left[x_m\right]f\left(x_m\right)$$

(32)

with $x>0$, where $I_{\left(M\right)}^{<1>,x}\left[u\right]f\left(u\right)=I_{\left(M\right)}^x\left[u\right]f\left(u\right)$.

The m-fold sequential GFD of the Riemann–Liouville type is defined as a composition of m GFDs with the kernel $K\left(x\right)$ in the form

$$D_{\left(K\right)}^{<m>,x}\left[u\right]f\left(u\right):=D_{\left(K\right)}^x\left[x_1\right]\dots D_{\left(K\right)}^{x_{m-1}}\left[x_m\right]f\left(x_m\right)$$

(33)

with $x>0$, where $D_{\left(K\right)}^{<1>,x}\left[u\right]f\left(u\right)=D_{\left(K\right)}^x\left[u\right]f\left(u\right)$.

Theorem 6.

The triple ${\mathcal{R}}_{-1}=(C_{-1}(0,\infty ),+,\ast )$ with standard addition + and multiplication * in form of the Laplace convolution is a commutative ring without divisors of zero.

Proof. 

Theorem 6 is proved in [62]. □

As a corollary of Theorem 6, one can state that the kernel $M^{<m>}\left(x\right)$ with $m\in \mathbb{N}$ belongs to the space $C_{-1}(0,\infty )$ since the kernel $M\left(x\right)$ belongs to the space $C_{-1}(0,\infty )$. Then, the m-fold sequential GFI can be represented as a GFI with the kernel $M^{<m>}\left(x\right)$ in the form

$$I_{\left(M\right)}^{<m>,x}\left[u\right]f\left(u\right)=(M^{<m>}\ast f)\left(x\right)=I_{\left(M^{<m>}\right)}^x\left[u\right]f\left(u\right),$$

(34)

where $x>0$.

The m-fold sequential GFD (33) is a generalization of the sequential GFD to the case of kernels that belong to the Luchko set ${\mathcal{L}}_n$.

Theorem 7

(First Fundamental Theorem of GFC for m-Fold Sequential GFD of Arbitrary Order). Let a kernel pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$.

Then, the m-fold sequential GFD (33) of the Riemann–Liouville type is a left inverse operator to the m-fold sequential GFI (32) in the form

$$D_{\left(K\right)}^{<m>,x}\left[u\right]I_{\left(M\right)}^{<m>,u}\left[w\right]f\left(w\right)=f\left(x\right)$$

(35)

for all $x>0$, if $f\left(x\right)\in C_{-1}(0,\infty )$ and $x>0$.

Proof. 

Theorem 7 is proved in the same manner as Theorem 4 in [31], p. 10. By repeated application of the first fundamental theorem of the GFD of an arbitrary order in the form of Theorem 3 for the GFI (13) and the GFD (14), one can obtain (35). □

Let us define the space $C_{-1,\left(M\right)}^m(0,\infty )$.

Definition 9.

Let the kernel pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$, and let the function $F\left(x\right)$ be represented in the form

$$F\left(x\right)=I_{\left(M\right)}^{<m>,x}\left[u\right]f\left(u\right),$$

(36)

for all $x>0$, where $f\left(x\right)\in C_{-1}(0,\infty )$.

Then, this set of functions, $F\left(x\right)$, is denoted as $C_{-1,\left(M\right)}^m(0,\infty )$.

For the space $C_{-1,\left(M\right)}^m(0,\infty )$, the m-fold sequential GFD (33) of an arbitrary order is a right inverse operator to the m-fold GFI (32).

Theorem 8

(Second Fundamental Theorem of GFC for m-Fold Sequential GFD of Arbitrary Order for $C_{-1,\left(M\right)}^m(0,\infty )$). Let the kernel pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$.

Then, the m-fold sequential GFD (33) of the Riemann–Liouville type is a right inverse operator to the m-fold GFI (32) in the form

$$I_{\left(M\right)}^{<m>,x}\left[u\right]D_{\left(K\right)}^{<m>,u}\left[w\right]F\left(w\right)=F\left(x\right),$$

(37)

for all $x>0$, if $F\left(x\right)\in C_{-1,\left(M\right)}^m(0,\infty )$.

Proof. 

Theorem 8 is proved in the same manner as Theorem 5 in [31], p. 10. □

Theorem 9.

Let the kernel pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$.

Then, the m-fold sequential GFD (33) of the Riemann–Liouville type can be represented as a GFD with the kernel $K^{<m>}\left(x\right)$ in the form

$$D_{\left(K\right)}^{<m>,x}\left[u\right]F\left(u\right)=D_{\left(K^{<m>}\right)}^x\left[u\right]F\left(u\right)$$

(38)

for $x>0$, if $F\in C_{-1,\left(M\right)}^m(0,\infty )$.

Proof. 

Theorem 9 is proved in [31], (see Theorem 5, p. 10). □

Remark 2.

Note that Equation (37) is not satisfied for the m-fold sequential GFD (33) defined on its domain

$$C_{-1,\left(K\right)}^{\left(m\right)}(0,\infty )=\{F\left(x\right)\in C_{-1}(0,\infty ):D_{\left(K\right)}^{<k>,x}\left[u\right]F\left(u\right)\in C_{-1}(0,\infty ),k=1,\dots ,m\}.$$

(39)

For the pair of the kernels $(M,K)\in {\mathcal{L}}_n$, $n\in \mathbb{N}$, the space $C_{-1,\left(M\right)}^m(0,\infty )$, which is defined by Definition 9, is a subspace of the space $C_{-1,\left(K\right)}^{\left(m\right)}(0,\infty )$ due to Theorem 7.

For the space $C_{-1,\left(K\right)}^{\left(m\right)}(0,\infty )$, a generalization of Equation (38) is considered to be the same as Theorem 6 in [31], p. 11, where a projector operator is used to prove the second fundamental theorem of the GFC of an m-fold sequential GFD of an arbitrary order.

3. Multi-Kernel GFC of Arbitrary Order

In this section, a multi-kernel approach, which is proposed in [35], is used to generalize the GFC of the sequential n-fold GF integrals and GF derivatives. The multi-kernel approach is used to expand the general fractional calculus due to its simultaneous use of different operator kernels. The multi-kernel approach is presented as a generalization of the Luchko results, which are derived in [31], for a set of kernel pairs $(M_j,K_j)\in {\mathcal{L}}_n$ with $n\in \mathbb{N}$, where $j=1,\dots ,m$. The multi-kernel general fractional operators of an arbitrary order are defined. The fundamental theorems of the multi-kernel GFC of an arbitrary order are proved.

Let us define the convolutional product $f^{<1|m>}\left(x\right)$ with $m\in \mathbb{N}$ by the equation

$$(f_1\ast \dots \ast f_m)\left(x\right),$$

(40)

where $f_j\left(x\right)\in C_{-1}(0,\infty )$ for all $j=1,\dots ,m$.

Using Theorem 6, one can state that if $f_j\left(x\right)\in C_{-1}(0,\infty )$ for all $j=1,\dots ,m$, then the convolutional product $f^{<1|m>}\left(x\right)$ with $m\in \mathbb{N}$, which is defined by Equation (40), also belongs to the space, i.e., $f^{<1|m>}\left(x\right)\in C_{-1}(0,\infty )$.

Let us define the multi-kernel m-fold sequential GFI.

Definition 10.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

The multi-kernel m-fold sequential GFI is defined as a sequential action of the GFIs $I_{\left(M_j\right)}^x\left[u_j\right]$ of an arbitrary order n with the kernels $M_j\left(x\right)$ and $j=1,\dots ,m$, in the form

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right):=I_{\left(M_1\right)}^x\left[x_2\right]\dots I_{\left(M_m\right)}^{x_m}\left[u\right]f\left(u\right)$$

(41)

with $x>0$ and ${\mathbb{I}}_{\left(M\right)}^{<1|1>,x}\left[u\right]f\left(u\right)=I_{\left(M_1\right)}^x\left[u\right]f\left(u\right)$, where the GFIs $I_{\left(M_j\right)}^x\left[u_j\right]$ are defined by Equation (13).

For the reverse order of kernels, the multi-kernel m-fold sequential GFI is defined as a composition of the GFIs $I_{\left(M_j\right)}^x\left[u_j\right]$ of an arbitrary order n with the kernels $M_j\left(x\right)$ and $j=1,\dots ,m$, in the form

$${\mathbb{I}}_{\left(M\right)}^{<m|1>,x}\left[u\right]f\left(u\right):=I_{\left(M_m\right)}^x\left[x_{m-1}\right]\dots I_{\left(M_1\right)}^{x_1}\left[u\right]f\left(u\right)$$

(42)

with $x>0$, where the GFIs $I_{\left(M_j\right)}^x\left[u_j\right]$ are defined by Equation (13).

The following statement describes the property of the m-fold sequential GFI for the multi-kernel case.

Theorem 10.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

Then, the multi-kernel GFI (41) of an arbitrary order can be represented as a GFI with the kernel $M^{<1|m>}\left(x\right)$ in the form

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)=I_{\left(M^{<1|m>}\right)}^x\left[u\right]f\left(u\right),$$

(43)

for $x>0$, if $f\left(x\right)\in C_{-1}(0,\infty )$.

Proof. 

Theorem 6, which is proved in [62], states that the triple ${\mathcal{R}}_{-1}=(C_{-1}(0,\infty ),+,\ast )$ is a commutative ring without divisors of zero. As a corollary of Theorem 6, one can state that the kernel $M^{<1|m>}\left(x\right)$ with $m\in \mathbb{N}$ belongs to the space $C_{-1}(0,\infty )$ since $M_j\left(x\right)$ belongs to the space $C_{-1}(0,\infty )$ for all $j=1,\dots ,m$, where $M_j\left(x\right)$ are the kernels of the GFIs of arbitrary orders. Then, the m-fold sequential GFI can be represented as a GFI with the kernel $M^{<1|m>}\left(x\right)$ in the form

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)=(M^{<1|m>}\ast f)\left(x\right)=I_{\left(M^{<1|m>}\right)}^x\left[u\right]f\left(u\right),$$

(44)

where $x>0$. □

As a result, using the commutative and associative properties of the Laplace convolution, one can obtain that, for general fractional integrals, the following equality is satisfied

$${\mathbb{I}}_{\left(M\right)}^{<m|1>,x}\left[u\right]f\left(u\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right).$$

(45)

For the case $m=3$, equality (45) means

$$I_{\left(M_3\right)}^x\left[x_2\right]I_{\left(M_2\right)}^{x_2}\left[x_1\right]I_{\left(M_1\right)}^{x_1}\left[u\right]f\left(u\right)=I_{\left(M_1\right)}^x\left[x_2\right]I_{\left(M_2\right)}^{x_2}\left[x_3\right]I_{\left(M_3\right)}^{x_3}\left[u\right]f\left(u\right).$$

(46)

The m-fold sequential GFD (33) is a generalization of the sequential GFD to the case of the kernels that belong to the Luchko set ${\mathcal{L}}_n$.

Let us define the multi-kernel m-fold sequential GFD of the Riemann–Liouville type.

Definition 11.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

The multi-kernel m-fold sequential GFD of the Riemann–Liouville type is defined as a composition of the GFIs $D_{\left(K_j\right)}^x\left[u_j\right]$ of an arbitrary order n with the kernels $K_j\left(x\right)$ and $j=1,\dots ,m$, in the form

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]f\left(u\right):=D_{\left(K_1\right)}^x\left[x_2\right]\dots D_{\left(K_m\right)}^{x_m}\left[u\right]f\left(u\right)$$

(47)

with $x>0$ and ${\mathbb{D}}_{\left(K\right)}^{<1|1>,x}\left[u\right]f\left(u\right)=D_{\left(K_1\right)}^x\left[u\right]f\left(u\right)$, where the GFDs $D_{\left(K_j\right)}^x\left[u_j\right]$ are defined by Equation (14).

For the reverse order of kernels, the multi-kernel m-fold sequential GFD of the Riemann–Liouville type is defined as a composition of the GFIs $D_{\left(K_j\right)}^x\left[u_j\right]$ of an arbitrary order n with the kernels $K_j\left(x\right)$ and $j=1,\dots ,m$, in the form

$${\mathbb{D}}_{\left(K\right)}^{<m|1>,x}\left[u\right]f\left(u\right):=D_{\left(K_m\right)}^x\left[x_{m-1}\right]\dots D_{\left(K_1\right)}^{x_1}\left[u\right]f\left(u\right)$$

(48)

with $x>0$, where the GFDs $D_{\left(K_j\right)}^x\left[u_j\right]$ are defined by Equation (14).

Remark 3.

The multi-kernel m-fold sequential GFD of the Caputo type is defined similarly by using the GFD (15) of the Caputo type with the kernel $K_j\left(x\right)$ and $j=1,\dots ,m$.

For example, Equations (47) and (48) with $m=3$ have the form

$${\mathbb{D}}_{\left(K\right)}^{<1|3>,x}\left[u\right]f\left(u\right)=D_{\left(K_1\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_3\right]D_{\left(K_3\right)}^{x_3}\left[u\right]f\left(u\right),$$

(49)

$${\mathbb{D}}_{\left(K\right)}^{<3|1>,x}\left[u\right]f\left(u\right)=D_{\left(K_3\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_1\right]D_{\left(K_1\right)}^{x_1}\left[u\right]f\left(u\right).$$

(50)

Remark 4.

It should be emphasized that, in contrast to general fractional integrals, for general fractional derivatives, the following inequality is satisfied

$${\mathbb{D}}_{\left(K\right)}^{<m|1>,x}\left[u\right]f\left(u\right)\ne {\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]f\left(u\right),$$

(51)

in the general case. For the case $m=2$, inequality (51) means

$${\mathbb{D}}_{\left(K\right)}^{<1|2>,x}\left[u\right]f\left(u\right)=D_{\left(K_1\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[u\right]f\left(u\right)\ne D_{\left(K_3\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[u\right]f\left(u\right)={\mathbb{D}}_{\left(K\right)}^{<2|1>,x}\left[u\right]f\left(u\right).$$

(52)

For the case $m=3$, equality (51) means

$$D_{\left(K_1\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_3\right]D_{\left(K_3\right)}^{x_3}\left[u\right]f\left(u\right)\ne D_{\left(K_3\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_1\right]D_{\left(K_1\right)}^{x_1}\left[u\right]f\left(u\right).$$

(53)

Let us prove the following fundamental theorem of the multi-kernel GFC.

Theorem 11

(First Fundamental Theorem of Multi-Kernel GFC of Arbitrary Order). Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

Then, the m-fold sequential GFD (47) of the Riemann–Liouville type is a left inverse operator to the m-fold sequential GFI (41) in the form

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,u}\left[w\right]f\left(w\right)=f\left(x\right)$$

(54)

for all $x>0$, if the function $f\left(x\right)$ belongs to the space $C_{-1}(0,\infty )$.

Proof. 

By repeated application of the first fundamental theorem of the GFD of an arbitrary order in the form of Theorem 3 for the GFI (41) and the GFD (47), one can obtain (54).

In order to explain further transformations of the general case, we first consider the transformation of the right-hand side of Equation (54) for the case $m=3$:

$${\mathbb{D}}_{\left(K\right)}^{<1|3>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|3>,u}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_3\right]D_{\left(K_3\right)}^{x_3}\left[u\right]I_{\left(M_1\right)}^u\left[x_2\right]I_{\left(M_2\right)}^{x_2}\left[x_3\right]I_{\left(M_3\right)}^{x_3}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_3\right]D_{\left(K_3\right)}^{x_3}\left[u\right]I_{\left(M_3\right)}^u\left[x_2\right]I_{\left(M_2\right)}^{x_2}\left[x_1\right]I_{\left(M_1\right)}^{x_1}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]D_{\left(K_2\right)}^{x_2}\left[x_3\right]I_{\left(M_2\right)}^{x_3}\left[x_1\right]I_{\left(M_1\right)}^{x_1}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]I_{\left(M_1\right)}^{x_2}\left[w\right]f\left(w\right)=f\left(x\right).$$

Here, Theorem 3, which is the first fundamental theorem of GFC, is used by taking into account that $f\left(x\right)\in C_{-1}(0,\infty )$.

Let us consider the expression ${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,u}\left[w\right]f\left(w\right)$ with $m\in \mathbb{N}$ in the general form

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,u}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]\dots D_{\left(K_{m-1}\right)}^{x_{m-1}}\left[x_m\right]D_{\left(K_m\right)}^{x_m}\left[u\right]I_{\left(M_1\right)}^u\left[x_2\right]I_{\left(M_2\right)}^{x_2}\left[x_3\right]\dots I_{\left(M_m\right)}^{x_m}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]\dots D_{\left(K_{m-1}\right)}^{x_{m-1}}\left[x_m\right]D_{\left(K_m\right)}^{x_m}\left[u\right]I_{\left(M_m\right)}^u\left[x_{m-1}\right]I_{\left(M_{m-1}\right)}^{x_{m-1}}\left[x_{m-2}\right]\dots I_{\left(M_1\right)}^{x_1}\left[w\right]f\left(w\right)=$$

$$D_{\left(K_1\right)}^x\left[x_2\right]\dots D_{\left(K_{m-1}\right)}^{x_{m-1}}\left[x_m\right]I_{\left(M_{m-1}\right)}^{x_m}\left[x_{m-2}\right]\dots I_{\left(M_1\right)}^{x_1}\left[w\right]f\left(w\right)=\dots =$$

$$D_{\left(K_1\right)}^x\left[x_2\right]I^{x_2}\left(M_1\right)\left[w\right]f\left(w\right)=f\left(x\right).$$

(55)

As a result, one can obtain

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,u}\left[w\right]f\left(w\right)=f\left(x\right).$$

(56)

□

To consider the second fundamental theorem of the multi-kernel GFC of an arbitrary order, one should define the set $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$.

Definition 12.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$, and let the function $F\left(x\right)$ be represented in the form

$$F\left(x\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)$$

(57)

for all $x>0$, where $f\left(x\right)\in C_{-1}(0,\infty )$.

Then, this set of functions, $F\left(x\right)$, is denoted as $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$.

Equation (57) means that

$$F\left(x\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)=I_{\left(M_1\right)}^x\left[x_1\right]\dots I_{\left(M_m\right)}^{x_{m-1}}\left[u\right]f\left(u\right).$$

(58)

Using property (45) in the form

$$I_{\left(M_1\right)}^x\left[x_2\right]\dots I_{\left(M_m\right)}^{x_m}\left[u\right]f\left(u\right)=I_{\left(M_m\right)}^x\left[x_{m-1}\right]\dots I_{\left(M_1\right)}^{x_1}\left[u\right]f\left(u\right)$$

(59)

Equation (58) can also be written as

$$F\left(x\right)=I_{\left(M_m\right)}^x\left[x_{m-1}\right]\dots I_{\left(M_1\right)}^{x_1}\left[u\right]f\left(u\right)={\mathbb{I}}_{\left(M\right)}^{<m|1>,x}\left[u\right]f\left(u\right).$$

(60)

For the set $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$, the m-fold sequential GFD (47) of an arbitrary order is also a right inverse operator to the m-fold GFI (41).

Theorem 12

(Second Fundamental Theorem of Multi-Kernel GFC of Arbitrary Order for $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$). Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

Then, the m-fold sequential GFD (47) of the Riemann–Liouville type is a right inverse operator to the m-fold GFI (32) in the form

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]F\left(w\right)=F\left(x\right)$$

(61)

for all $x>0$, if the function $F\left(x\right)$ belongs to the set $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$.

Proof. 

For the case $M_j\left(x\right)=M\left(x\right)$ for all $j=1,\dots ,m$, Theorem 8 is proved in [31]. Let us consider the multi-kernel case.

Using the premise that $F\left(x\right)$ belongs to the set $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$, one can obtain

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]F\left(w\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,w}\left[s\right]f\left(s\right).$$

(62)

Let us consider the expression ${\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,w}\left[s\right]f\left(s\right)$ in the general form

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,w}\left[s\right]f\left(s\right)=$$

$$D_{\left(K_1\right)}^u\left[x_2\right]\dots D_{\left(K_{m-1}\right)}^{x_{m-1}}\left[x_m\right]D_{\left(K_m\right)}^{x_m}\left[w\right]I_{\left(M_1\right)}^w\left[x_2\right]I_{\left(M_2\right)}^{x_2}\left[x_3\right]\dots I_{\left(M_m\right)}^{x_m}\left[s\right]f\left(s\right)=$$

$$D_{\left(K_1\right)}^u\left[x_2\right]\dots D_{\left(K_{m-1}\right)}^{x_{m-1}}\left[x_m\right]D_{\left(K_m\right)}^{x_m}\left[w\right]I_{\left(M_m\right)}^w\left[x_{m-1}\right]I_{\left(M_{m-1}\right)}^{x_{m-1}}\left[x_{m-2}\right]\dots I_{\left(M_1\right)}^{x_1}\left[s\right]f\left(s\right)=$$

$$D_{\left(K_1\right)}^u\left[x_2\right]\dots D_{\left(K_{m-1}\right)}^{x_{m-1}}\left[x_m\right]I_{\left(M_{m-1}\right)}^{x_m}\left[x_{m-2}\right]\dots I_{\left(M_1\right)}^{x_1}\left[s\right]f\left(s\right)=\dots =$$

$$D_{\left(K_1\right)}^u\left[x_2\right]I_{\left(M_1\right)}^{x_2}\left[s\right]f\left(s\right)=f\left(u\right).$$

(63)

Here, the following property of the GFIs is used

$${\mathbb{I}}_{\left(M\right)}^{<m|1>,x}\left[u\right]F\left(u\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]F\left(u\right).$$

(64)

As a result, Equation (64) has the form

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,w}\left[s\right]f\left(s\right)=f\left(u\right).$$

(65)

Using equality (65), Equation (62) gives

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]F\left(w\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right).$$

(66)

Then, using the premise that the function $f\left(x\right)$ belongs to the space $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$ and it is represented as (57), Equation (66) gives

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right)}^{<1|m>,u}\left[w\right]F\left(w\right)=F\left(x\right).$$

(67)

This proves the equality. □

The following statement describes the property of the m-fold sequential GFD for the multi-kernel case.

Theorem 13.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

Then, the multi-kernel GFD (47) of an arbitrary order can be represented as a GFD with the kernel $K^{<1|m>}\left(x\right)$ in the form

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]F\left(u\right)=D_{\left(K^{<1|m>}\right)}^x\left[u\right]F\left(u\right),$$

(68)

for $x>0$, if $F\left(x\right)\in C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$.

Proof. 

For the function $F\left(x\right)$ from $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$, using the first fundamental theorem of a multi-kernel GFD of arbitrary order, one can obtain

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]F\left(u\right)={\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,u}\left[w\right]f\left(w\right)={\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<m|1>,u}\left[w\right]f\left(w\right)=f\left(x\right)=$$

$$D_{\left(K^{<1|m>}\right)}^x\left[u\right]I_{\left(M^{<m|1>}\right)}^u\left[w\right]f\left(w\right)=D_{\left(K^{<1|m>}\right)}^x\left[u\right]I_{\left(M^{<1|m>}\right)}^u\left[w\right]f\left(w\right)=$$

$$D_{\left(K^{<1|m>}\right)}^x\left[u\right]{\mathbb{I}}_{\left(M\right)}^{<1|m>,u}\left[w\right]f\left(w\right)=D_{\left(K^{<m>}\right)}^x\left[u\right]F\left(u\right),$$

where $x>0$. □

4. General Fractional Calculus on Finite Interval [a, b]

The formulation of the GFC of an arbitrary order, which was proposed in the previous section for the positive semi-axis $(0,\infty )$, can be generalized to intervals including the negative part of the real axis.

Let us first give some comments about fractional integral with a power-law kernel for finite intervals. In this case, the fractional calculus is based on the Riemann–Liouville fractional integrals and the Riemann–Liouville and Caputo fractional derivatives. The Riemann–Liouville fractional integral on the finite interval $(a,b)$ is defined by the equation

$$\left(I_{a+}^{\alpha }g\right)\left(x\right)={\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}g\left(u\right)du,$$

(69)

where $a<x<b$ and $\alpha >0$. The fractional integral (69) is defined for the functions $g\left(x\right)\in L_1(a,b)$.

Using Table 9.1 of [1], p. 173, one can see that the Riemann–Liouville fractional integrals on the finite interval $(a,b)$ are considered for functions of the form

$$g\left(x\right)=f(x-a).$$

(70)

Almost all equations in Table 9.1 in [1] have the form

$${\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}f(u-a)du=G(x-a).$$

(71)

For example, the first equation in Table 9.1 in [1], p. 173, has the form

$${\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{(u-a)}^{\beta -1}du=\frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}{(x-a)}^{\beta -1}.$$

(72)

For another example, Equation (23) in Table 9.1 in [1], p. 173, has the form

$${\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{(u-a)}^{\beta -1}{E}_{\mu ,\beta }\left[{(u-a)}^{\mu }\right]du={(x-a)}^{\alpha +\beta -1}{E}_{\mu ,\alpha +\beta }\left[{(x-a)}^{\mu }\right],$$

(73)

where $\alpha >0$, $\beta >0$, $\mu >0$.

As a result, to consider the Riemann–Liouville fractional integrals (69) on the finite interval $(a,b)$, one can use the equation

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)={\int }_a^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}f(u-a)du.$$

(74)

The representations of type (74) can be used to extend the applications of the general fractional calculus, in which an interval $[0,\infty )$ is considered. In this extension of the GFC, one can consider finite intervals $[a,b]$, where $-\infty <a<b<\infty$, or infinite intervals $[a,b)$, where $-\infty <a<b=\infty$.

Definition 13.

Let the kernel pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$, and let the function $f\left(x\right)$ belong to the space $C_{-1}(0,\infty )$. Let $-\infty <a<b\le \infty$.

Then, the GFI $I_{\left(M\right),a+}^x$ on the interval $(a,b)$ is defined by the equation

$$I_{\left(M\right),a+}^x\left[t\right]f(t-a):={\int }_a^{x}M(x-t)f(t-a)dt,$$

(75)

where $x\ge t>a$.

Then, the GFD $D_{\left(K\right),a+}^x$ on the interval $(a,b)$ is defined by the equation

$$D_{\left(K\right),a+}^x\left[t\right]f(t-a):=\frac{d^n}{d{x}^n}{\int }_a^{x}K(x-t)f(t-a)dt,$$

(76)

where $x\ge t>a$.

Let us prove the following property.

Proposition 1.

Let the kernel pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_n$, and let the function $f\left(x\right)$ belong to the space $C_{-1}(0,\infty )$. Let $-\infty <a<b\le \infty$.

Then, the GFI $I_{\left(M\right),a+}^x$ on the interval $(a,b)$ can be represented as

$$I_{\left(M\right),a+}^x\left[t\right]f(t-a)=I_{\left(M\right)}^{x-a}\left[u\right]f\left(u\right),$$

(77)

where $x\ge t>a$ and $x-a\ge u>0$.

Then, the GFD $D_{\left(M\right),a+}^x$ on the interval $(a,b)$ can be represented as

$$D_{\left(M\right),a+}^x\left[t\right]f(t-a)=D_{\left(M\right)}^{x-a}\left[u\right]f\left(u\right),$$

(78)

where $x\ge t>a$ and $x-a\ge u>0$.

Proof. 

Let us consider the GF integral

$$I_{\left(M\right),a+}^x\left[t\right]g\left(t\right):={\int }_a^{x}M(x-t)g\left(t\right)dt,$$

(79)

where $x\ge t>a$. Then, Equation (79) with $g\left(t\right)=f(t-a)$ has the form

$$I_{\left(M\right),a+}^x\left[t\right]f(t-a)={\int }_a^{x}M(x-t)f(t-a)dt.$$

(80)

Using the variable $u=t-a$, one can obtain

$$I_{\left(M\right),a+}^x\left[t\right]f(t-a)={\int }_0^{x-a}M(x-(u+a))f\left(u\right)du={\int }_0^{x-a}M((x-a)-u)f\left(u\right)du.$$

(81)

Therefore, Equation (81) can be written as

$$I_{\left(M\right),a+}^x\left[t\right]f(t-a)=I_{\left(M\right),0+}^{x-a}\left[u\right]f\left(u\right)=I_{\left(M\right)}^{x-a}\left[u\right]f\left(u\right),$$

(82)

where $x\ge t>a$ and $x-a\ge u>0$. Here, $f\left(x\right)\in C_{-1}(0,\infty )$ is defined on the interval $[0,\infty )$ or $[0,b-a]$ with $-\infty <a<b\le \infty$. As a result, Equation (77) was proved.

For the GFD $D_{\left(M\right),a+}^x$, Equation (78) is proved similarly. □

Equation (77) expresses the GF integral $I_{\left(M\right),a+}^x$ on the interval $(a,b)$ in terms of the GF integral $I_{\left(M\right)}^x$ by Equation (77), where the function $f\left(x\right)\in C_{-1}(0,\infty )$ can be considered for $[0,b-a]$.

Similarly, one can define multi-kernel GF integrals on the finite interval $(a,b)$.

Definition 14.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$. Let the function $f\left(x\right)$ belong to the space $C_{-1}(0,\infty )$, and let $-\infty <a<b\le \infty$.

The multi-kernel m-fold sequential GFI on the interval $(a,b)$ is defined as a composition of m GFIs with the kernels $M_j\left(x\right)$, $j=1,\dots ,m$, in the form

$$\left({\mathbb{I}}_{\left(M\right),a+}^{<1|m>}f\right)\left(x\right)={\mathbb{I}}_{\left(M\right),a+}^{<1|m>,x}\left[u\right]f(u-a):=I_{\left(M_1\right),a+}^x\left[x_2\right]\dots I_{\left(M_m\right),a+}^{x_m}\left[u\right]f(u-a),$$

(83)

where $x\ge t>a$ and ${\mathbb{I}}_{\left(M\right),a+}^{<1|1>,x}\left[u\right]f(u-a)=I_{\left(M_1\right),a+}^x\left[u\right]f(u-a)$.

The multi-kernel m-fold sequential GFD on the interval $(a,b)$ is defined as a composition of m GFDs with the kernels $K_j\left(x\right)$, $j=1,\dots ,m$, in the form

$$\left({\mathbb{D}}_{\left(K\right),a+}^{<1|m>}f\right)\left(x\right)={\mathbb{D}}_{\left(K\right),a+}^{<1|m>,x}\left[u\right]f(u-a):=D_{\left(K_1\right),a+}^x\left[x_2\right]\dots D_{\left(K_m\right),a+}^{x_m}\left[u\right]f(u-a),$$

(84)

where $x\ge t>a$ and ${\mathbb{D}}_{\left(K\right),a+}^{<1|1>,x}\left[u\right]f(u-a)=D_{\left(K_1\right),a+}^x\left[u\right]f(u-a)$.

Using Proposition 1, one can prove the following statement.

Theorem 14.

Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $n\in \mathbb{N}$. Let the function $f\left(x\right)$ belong to the space $C_{-1}(0,\infty )$, and let $-\infty <a<b\le \infty$.

The multi-kernel m-fold sequential GFI on the interval $(a,b)$ can be represented as

$${\mathbb{I}}_{\left(M\right),a+}^{<1|m>,x}\left[u\right]f(u-a)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x-a}\left[w\right]f\left(w\right)$$

(85)

with $x>a$ and ${\mathbb{I}}_{\left(M\right),a+}^{<1|1>,x}\left[u\right]f(u-a)=I_{\left(M_1\right),a+}^x\left[u\right]f(u-a)$, where $x\ge t>a$ and $x-a\ge u>0$.

The multi-kernel m-fold sequential GFD on the interval $(a,b)$ can be represented as

$${\mathbb{D}}_{\left(K\right),a+}^{<1|m>,x}\left[u\right]f(u-a)={\mathbb{D}}_{\left(K\right)}^{<1|m>,x-a}\left[w\right]f\left(w\right).$$

(86)

with $x>a$ and ${\mathbb{D}}_{\left(K\right),a+}^{<1|1>,x}\left[u\right]f(u-a)=D_{\left(K_1\right),a+}^x\left[u\right]f(u-a)$, where $x\ge t>a$ and $x-a\ge u>0$.

Proof. 

By repeated application of Proposition 1 to Equation (83), one can obtain

$${\mathbb{I}}_{\left(M\right),a+}^{<1|m>,x}\left[u\right]f(u-a):=I_{\left(M_1\right),a+}^x\left[x_2\right]I_{\left(M_2\right),a+}^{x_2}\left[x_3\right]\dots I_{\left(M_{m-1}\right),a+}^{x_{m-1}}\left[x_m\right]I_{\left(M_m\right),a+}^{x_m}\left[u\right]f(u-a)=$$

$$I_{\left(M_1\right),a+}^x\left[x_2\right]I_{\left(M_2\right),a+}^{x_2}\left[x_3\right]\dots I_{\left(M_{m-1}\right),a+}^{x_{m-1}}\left[x_m\right]I_{\left(M_m\right),0+}^{x_m-a}\left[w\right]f\left(w\right)=$$

$$I_{\left(M_1\right),a+}^x\left[x_2\right]I_{\left(M_2\right),a+}^{x_2}\left[x_3\right]\dots I_{\left(M_{m-1}\right),0+}^{x_{m-1}-a}\left[w_m\right]I_{\left(M_m\right),0+}^{w_m}\left[w\right]f\left(w\right)=$$

$$I_{\left(M_1\right),a+}^x\left[x_2\right]I_{\left(M_2\right),a+}^{x_2}\left[x_3\right]\dots I_{\left(M_{m-1}\right),0+}^{w_{m-1}}\left[w_m\right]I_{\left(M_m\right),0+}^{w_m}\left[w\right]f\left(w\right)=$$

$$I_{\left(M_1\right),a+}^x\left[x_2\right]I_{\left(M_2\right),0+}^{x_2-a}\left[w_3\right]\dots I_{\left(M_{m-1}\right),0+}^{w_{m-1}}\left[w_m\right]I_{\left(M_m\right),0+}^{w_m}\left[w\right]f\left(w\right)=$$

$$I_{\left(M_1\right),0+}^{x-a}\left[w_2\right]I_{\left(M_2\right),0+}^{w_2}\left[w_3\right]\dots I_{\left(M_{m-1}\right),0+}^{w_{m-1}}\left[w_m\right]I_{\left(M_m\right),0+}^{w_m}\left[w\right]f\left(w\right)=$$

$${\mathbb{I}}_{\left(M\right),0+}^{<1|m>,x-a}\left[w\right]f\left(w\right)={\mathbb{I}}_{\left(M\right)}^{<1|m>,x-a}\left[w\right]f\left(w\right).$$

(87)

This proves the equality. □

As a result, to consider the general fractional calculus on a finite interval $(a,b)$, where $-\infty <a<b\le \infty$, one can use the already existing general fractional calculus in the Luchko form.

Theorem 15

(First Fundamental Theorem of Multi-Kernel GFC for $(a,\infty$). Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

Then, the m-fold sequential GFD (47) of the Riemann–Liouville type is a left inverse operator to the m-fold sequential GFI (41) in the form

$${\mathbb{D}}_{\left(K\right),a+}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right),a+}^{<1|m>,u}\left[t\right]f(t-a)=f(x-a)$$

(88)

for all $x>a$, if the function $f\left(x\right)$ belongs to the space $C_{-1}(0,\infty )$.

Proof. 

The proof of Theorem 15 is reduced to the proof of Theorem 11 by the following transformations

$${\mathbb{D}}_{\left(K\right),a+}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right),a+}^{<1|m>,u}\left[w\right]f(t-a)={\mathbb{D}}_{\left(K\right),a+}^{<1|m>,x}\left[u\right]{\mathbb{I}}_{\left(M\right),0+}^{<1|m>,u-a}\left[w\right]f\left(w\right)=$$

$${\mathbb{D}}_{\left(K\right),0+}^{<1|m>,x-a}\left[s\right]{\mathbb{I}}_{\left(M\right),0+}^{<1|m>,s}\left[w\right]f\left(w\right).$$

(89)

□

Theorem 16

(Second Fundamental Theorem of Multi-Kernel GFC for $(a,\infty )$ and $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$). Let the kernel pairs $(M_j,K_j)$ with $j=1,\dots ,m$ belong to the Luchko set ${\mathcal{L}}_n$ with $m,n\in \mathbb{N}$.

Then, the m-fold sequential GFD (47) of the Riemann–Liouville type is a right inverse operator to the m-fold GFI (32) in the form

$${\mathbb{I}}_{(M,a+)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right),a+}^{<1|m>,u}\left[t\right]F(t-a)=F(x-a)$$

(90)

for all $x>a$, if the function $F\left(x\right)$ belongs to the set $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$.

Proof. 

The proof of Theorem 16 is reduced to the proof of Theorem 12 by the following transformations

$${\mathbb{I}}_{(M,a+)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right),a+}^{<1|m>,u}\left[t\right]F(t-a)={\mathbb{I}}_{(M,a+)}^{<1|m>,x}\left[u\right]{\mathbb{D}}_{\left(K\right),0+}^{<1|m>,u-a}\left[w\right]F\left(w\right)=$$

$${\mathbb{I}}_{(M,0+)}^{<1|m>,x-a}\left[s\right]{\mathbb{D}}_{\left(K\right),0+}^{<1|m>,s}\left[w\right]F\left(w\right).$$

(91)

□

5. Examples of Multi-Kernel GF Operators of Arbitrary Order

In this section, some examples of multi-kernel GF operators are considered.

5.1. Equations of Multi-Kernel GF Operators

Let us give some statements about the multi-kernel GFI and GFD.

Proposition 2.

Let the kernel pairs $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ for all $j=1,\dots ,m$ with $m\in \mathbb{N}$.

Then, the kernel pairs $(h_{n-1}\ast M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_n$ for all $j=1,\dots ,m$ with $m\in \mathbb{N}$ and all $n>1$, $n\in \mathbb{N}$. where

$$h_n\left(x\right)=\frac{x^{n-1}}{(n-1)!}.$$

(92)

and $x>0$.

Proof. 

The statement of Proposition 2 follows directly from Definitions 10 and 11 of the Luchko set ${\mathcal{L}}_n$. □

Note that function (92) is also denoted as ${\left\{1\right\}}^n$.

Proposition 3.

Let the kernel pairs $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ for all $j=1,\dots ,m$ with $m\in \mathbb{N}$.

Then, the kernels $K^{<1|m>}\left(x\right)$ and $M^{<1|m>}\left(x\right)$ can be represented in the following forms. For all $n\in \mathbb{N}$, the kernel $K^{<1|m>}\left(x\right)$ is

$$K^{<1|m>}\left(x\right)=(K_1\ast \dots \ast K_m)\left(x\right).$$

(93)

For $n\in \mathbb{N}$, the kernel $M^{<1|m>}\left(x\right)$ is

$$M^{<1|m>}\left(x\right)=,\left\{\begin{array}{cc}(M_1\ast \dots \ast M_m)\left(x\right)\hfill & n=1,\hfill \\ (h_{m(n-1)}\ast M_1\ast \dots \ast M_m)\left(x\right)\hfill & n\ge 1.\hfill \end{array}\right.$$

(94)

Proof. 

The statement of Proposition 3 follows directly from Proposition 2, Definitions 10 and 11 of the multi-kernel GFI and GFD of an arbitrary order, and Theorems 10 and 13. □

Let us define a set of kernel pairs for the multi-kernel GFI and GFD of an arbitrary order.

Definition 15.

Let the kernel pairs $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ for all $j=1,\dots ,m$ with $m\in \mathbb{N}$.

Then, the set of kernel pairs $(M^{<1|m>},K^{<1|m>})$, in which the kernels can be represented in forms (93) and (94), is denoted as ${\mathcal{L}}_n^{<1|m>}$, where $n,m\in \mathbb{N}$.

Definition 16.

Let the pair $(M,K)$ belong to the Luchko set ${\mathcal{L}}_1$.

Then, a set of kernel pairs $(M^{<1|m>},K^{<1|m>})$, in which the kernels can be represented in forms (93) and (94) with $M_j\left(x\right)=M\left(x\right)$ and $K_j\left(x\right)=K\left(x\right)$ for all $j=1,\dots ,m$, is denoted as ${\mathcal{L}}_n^{<m>}$.

The set ${\mathcal{L}}_n^{<m>}$ is a subset of the set ${\mathcal{L}}_n^{<1|m>}$. For $m=1$, one can see that ${\mathcal{L}}_n^{<1|1>}={\mathcal{L}}_n$ and ${\mathcal{L}}_n^{<1>}={\mathcal{L}}_n$.

Theorem 17.

Let the kernel pair $(M^{<1|m>},K^{<1|m>})$ belong to the set ${\mathcal{L}}_n^{<1|m>}$ such that the kernels can be represented in forms (93) and (94), where the pairs $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ for all $j=1,\dots ,m$ with $m\in \mathbb{N}$.

Then, the multi-kernel GFI of an arbitrary order can be represented by the equations

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)={\int }_0^x{M}^{<1|m>}(x-u)f\left(u\right)du=$$

$${\int }_0^x(h_{m(n-1)}\ast M_1\ast \dots \ast M_m)(x-u)f\left(u\right)du,$$

(95)

if $f\left(x\right)\in C_{-1}(0,\infty )$ and $n>1$. For $n=1$, the multi-kernel GFI and GFD of an arbitrary order have the form

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)={\int }_0^x{M}^{<1|m>}(x-u)f\left(u\right)du={\int }_0^x(M_1\ast \dots \ast M_m)(x-u)f\left(u\right)du.$$

(96)

Then, the multi-kernel GFD of an arbitrary order can be represented by the equations

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]F\left(u\right)=\frac{d^{nm}}{d{x}^{nm}}{\int }_0^x{K}^{<1|m>}(x-u)F\left(u\right)du=$$

$$\frac{d^{nm}}{d{x}^{nm}}{\int }_0^x(K_1\ast \dots \ast K_m)(x-u)F\left(u\right)du,$$

(97)

if $F\left(x\right)\in C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$.

Proof. 

The proof of Theorem 17 is based on Theorems 10 and 13. Using Theorem 10, one can obtain

$${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\left[u\right]f\left(u\right)=I_{\left(M^{<1|m>}\right)}^x\left[u\right]f\left(u\right),$$

(98)

if $f\left(x\right)$ belongs to the set $C_{-1}(0,\infty )$. Using Theorem 13, one can obtain

$${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}\left[u\right]F\left(u\right)=D_{\left(K^{<1|m>}\right)}^x\left[u\right]F\left(u\right),$$

(99)

if $F\left(x\right)$ belongs to the set $C_{-1,\left(M\right)}^{<1|m>}(0,\infty )$. Then, using the definitions of the GFI $I_{\left(M\right)}^x$, and the GFD $D_{\left(K\right)}^x$, Equation (93) gives the representations (95) and (96) as well as Equation (97). □

Note that the kernels $M^{<1|m>}\left(x\right)$ are defined for the kernels from the Luchko set ${\mathcal{L}}_n$, where the kernel of the GFI contains the function ${\left\{1\right\}}^{n-1}=h_{n-1}\left(x\right)$ (see Theorem 2). Therefore, Equation (94) contains the function $h_{m(n-1)}$ since the kernels $M_j\left(x\right)$ belong to the Luchko set ${\mathcal{L}}_1$.

5.2. Dimensions of Multi-Kernel GFI and GFD

For applications, the standard physical dimensions of the variables and quantities should be preserved [48,61]. To preserve the standard physical dimensions, one can assume that the dimensions of the GFI and GFD must coincide with the dimensions of the standard derivative and integral of the integer orders

$$\left[I_{\left(M_j\right)}^x\left[u\right]f\left(u\right)\right]=\left[{\int }_0^{x}f\left(u\right)du\right],\left[D_{\left(K_j\right)}^x\left[u\right]F\left(u\right)\right]=\left[F^{\left(1\right)}\left(x\right)\right].$$

(100)

Equation (100) lead to the dimensions of the GFI and GFD kernels, which belong to the Luchko set ${\mathcal{L}}_1$, in the form

$$\left[K_j\right]={\left[x\right]}^{-1},\left[M_j\right]={\left[x\right]}^0,$$

(101)

where ${\left[x\right]}^0=\left[1\right]$ denotes the dimensionlessness of the quantity.

Using equalities (101), one can obtain the dimensions of the kernels, which belongs to the sets ${\mathcal{L}}_n$ and ${\mathcal{L}}_n^{<1|m>}$.

As a result, one can obtain the physical dimensions of the multi-kernel GFI ${\mathbb{I}}_{\left(M\right)}^{<1|m>,x}$ and the multi-kernel GFD ${\mathbb{D}}_{\left(K\right)}^{<1|m>,x}$. To preserve the standard physical dimensions, one can assume that the dimensions of the GFI and GFD must coincide with the dimensions of the standard derivative and integral of the integer orders in the form

$$\left[{\mathbb{I}}_{\left(M\right)}^{<1|m>,x}\right]=\left[I_{0+}^{nm}f\left(x\right)\right],\left[{\mathbb{D}}_{\left(K\right)}^{<1|m>,x}f\left(u\right)\right]=\left[f^{nm}\left(x\right)\right].$$

(102)

Using the following

$$\left(I_{0+}^{n}f\right)\left(x\right)={\int }_0^{x}d{u}_1{\int }_0^{u_1}d{u}_2\dots {\int }_0^{u_{n-1}}d{u}_{n}f\left(u_n\right)=$$

$${\int }_0^x\frac{{(x-u)}^{n-1}}{(n-1)!}f\left(u\right)du={\int }_0^x{h}_n(x-u)f\left(u\right)du,$$

(103)

one can obtain

$$\left[I_{0+}^n\right]={\left[x\right]}^n,\left[h_n\left(x\right)\right]={\left[x\right]}^{n-1}.$$

(104)

Then, using (94) for $n>1$, $n\in \mathbb{N}$, one can derive the dimension of the kernel $M^{<1|m>}\left(x\right)$ via the following transformations

$$\left[M^{<1|m>}\left(x\right)\right]=\left[(h_{m(n-1)}\ast M_1\ast \dots \ast M_m)\left(x\right)\right]=$$

$$\left[h_{m(n-1)}\left(x\right)\right]{\left({\left[x\right]}^0\right)}^m{\left[x\right]}^m={\left[x\right]}^{m(n-1)-1+m}={\left[x\right]}^{mn-1}.$$

(105)

Using Equation (93), one can obtain

$$\left[K^{<1|m>}\left(x\right)\right]=\left[(K_1\ast \dots \ast K_m)\left(x\right)\right]={\left({\left[x\right]}^{-1}\right)}^n{\left[x\right]}^{n-1}={\left[x\right]}^{-1}.$$

(106)

5.3. Examples of Kernel Pairs from Luchko Set ${\mathcal{L}}_1$

Let us give examples of the kernel pairs $(M_j\left(x\right),K_j\left(x\right))$ that belong to the Luchko set ${\mathcal{L}}_1$ and have the physical dimensions $\left[M_j\left(x\right)\right]={\left[x\right]}^0$ and $\left[K_j\left(x\right)\right]={\left[x\right]}^{-1}$, where $j\in \mathbb{N}$. In these examples, $\lambda >0$, $\left[\lambda \right]={\left[x\right]}^{-1}$, and $x>0$.

• The kernels of the power-law type have the form

  $$M_a\left(x\right)=h_{\alpha }\left(\lambda x\right)=\frac{{\left(\lambda x\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)},K_a\left(x\right)=\lambda h_{1-\alpha }\left(\lambda x\right)=\frac{\lambda {\left(\lambda x\right)}^{-\alpha }}{\Gamma (1-\alpha )}.$$

  (107)
• The kernels of the Gamma distribution type have the form

  $$M_b\left(x\right)=h_{\alpha ,\lambda }\left(\lambda x\right)=\frac{{\left(\lambda x\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{e}^{-\lambda x},K_b\left(x\right)=\lambda h_{1-\alpha ,\lambda }\left(\lambda x\right)+\frac{\lambda }{\Gamma (1-\alpha )}\gamma (1-\alpha ,\lambda x),$$

  (108)

  where $\gamma (\beta ,x)$ is the incomplete gamma function (see Section 9 in [63], pp. 134–142).
• The kernels of the Mittag–Leffler type have the form

  $$M_c\left(x\right)={\left(\lambda x\right)}^{\beta -1}{E}_{\alpha ,\beta }[-{\left(\lambda x\right)}^{\alpha }],K_c\left(x\right)=\lambda h_{\alpha -\beta ,\lambda }\left(\lambda x\right)+\lambda h_{1-\beta }\left(\lambda x\right),$$

  (109)

  where $0<\alpha \le \beta 1$, $E_{\alpha ,\beta }\left[z\right]$ is the two-parameter Mittag–Leffler function (see Section 3 in [64], pp. 17–54, [65] and Section 1.8 in [4], pp. 40–45). Here, the function

  $$e_{\alpha ,\beta }\left(x\right):=x^{\beta -1}{E}_{\alpha ,\beta }[-x^{\alpha }]$$

  (110)

  can be used to simplify notations. Using function (110), the kernel $M_c\left(x\right)$ can be written as

  $$M_c\left(x\right)=e_{\alpha ,\beta }\left(\lambda x\right).$$

  (111)
• The kernels of the Bessel type have the form

  $$M_d\left(x\right)={\left(\sqrt{\lambda x}\right)}^{\alpha -1}{J}_{\alpha -1}\left(2\sqrt{\lambda x}\right),K_d\left(x\right)=\lambda {\left(\sqrt{\lambda x}\right)}^{-\alpha }{I}_{-\alpha }\left(2\sqrt{\lambda x}\right).$$

  (112)

  where the function $J_{\alpha }\left(u\right)$ is the Bessel function, and $I_{\alpha }\left(u\right)$ is the modified Bessel function. The function

  $${\omega }_{\alpha }\left(x\right):={\left(\sqrt{x}\right)}^{\alpha -1}{J}_{\alpha -1}\left(2\sqrt{x}\right)$$

  (113)

  can be used to simplify notations. Using function (113), the kernel $M_d\left(x\right)$ can be written as

  $$M_d\left(x\right)={\omega }_{\alpha }\left(\lambda x\right).$$

  (114)
• The kernels of the hypergeometric Kummer type have the form

  $$M_f\left(x\right)={\left(\lambda x\right)}^{\beta -1}\Phi (\alpha ;\beta ;-\lambda x),K_e\left(x\right)=\frac{\lambda sin\left(\pi \beta \right)}{\pi }{\left(\lambda x\right)}^{-\beta }\Phi (-\alpha ;1-\beta ;-\lambda x),$$

  (115)

  where $\Phi (\alpha ;\beta ;x)$ is the confluent hypergeometric Kummer function (Section 1.6 in [4], pp. 29–30). In Equation (115), one can use the function

  $${\varphi }_{\alpha ,\beta }\left(x\right):=x^{\beta -1}\Phi (\alpha ;\beta ,-x)$$

  (116)

  to simplify notations. Using function (116), the kernel $M_f\left(x\right)$ can be written as

  $$M_f\left(x\right)={\varphi }_{\alpha ,\beta }\left(\lambda x\right).$$

  (117)

Remark 5.

For other examples of kernel pairs from the Luchko set ${\mathcal{L}}_1$, see article [48]. Note that the examples can be expanded by using kernel pairs of the form $(M_{j,new}={\lambda }^{-1}{K}_j\left(x\right),K_{j,new}=\lambda M_j\left(x\right))$ for each pair $(M_j\left(x\right),K_j\left(x\right))$ of these examples [48].

5.4. Examples of Calculating GF Integrals

To derive the multi-kernel GF operators, one should use equations of GF integration.

Proposition 4.

Equation

$$(h_{\alpha }\left(\lambda x\right)\ast h_{\beta }\left(\lambda x\right))={\lambda }^{-1}{h}_{\alpha +\beta }\left(\lambda x\right)$$

(118)

holds for all $x>0$, where $\alpha >0$, $\lambda >0$.

Proof. 

Using function (107), one can consider the equation

$${\int }_0^x{h}_{\alpha }(x-u)h_{\beta }\left(u\right)du=h_{\alpha +\beta }\left(x\right),$$

(119)

and the equation

$${\int }_0^x{h}_{\alpha }\left(\lambda (x-u)\right)h_{\beta }\left(\lambda u\right)du={\lambda }^{-1}{h}_{\alpha +\beta }\left(\lambda x\right),$$

(120)

where $\alpha >0$ and $\beta >0$. Equation (119) can be written in the form

$$(h_{\alpha }\ast h_{\beta })\left(x\right)=h_{\alpha +\beta }\left(x\right),$$

(121)

and Equation (120) as

$$(h_{\alpha }\left(\lambda x\right)\ast h_{\beta }\left(\lambda x\right))={\lambda }^{-1}{h}_{\alpha +\beta }\left(\lambda x\right).$$

(122)

□

Proposition 5.

Equation

$$(h_{\alpha ,\lambda }\left(\lambda x\right)\ast h_{\beta ,\lambda }\left(\lambda x\right))={\lambda }^{-1}{h}_{\alpha +\beta ,\lambda }\left(\lambda x\right)$$

(123)

holds for all $x>0$, where $\alpha >0$, $\lambda >0$.

Proof. 

For function (108), one can consider the equation

$${\int }_0^x{h}_{\alpha ,\lambda }(x-u)h_{\beta ,\lambda }\left(u\right)du=h_{\alpha +\beta ,\lambda }\left(x\right)$$

(124)

which can be written as

$$(h_{\alpha ,\lambda }\left(\lambda x\right)\ast h_{\beta ,\lambda }\left(\lambda x\right))={\lambda }^{-1}{h}_{\alpha +\beta ,\lambda }\left(\lambda x\right).$$

(125)

□

Proposition 6.

Equation

$$(h_{\mu }\left(\lambda x\right)\ast e_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-1}{e}_{\alpha ,\beta +\mu }\left(\lambda x\right)$$

(126)

holds for all $x>0$, where $\alpha >0$, $\beta >0$, $\mu >0$, $\lambda >0$.

Proof. 

For function (109), one can use Equation (4).4.5 of [64], p. 61, (see also Equation (23) in Table 9.1 in [1], p. 173) in the form

$$\frac{1}{\Gamma \left(\mu \right)}{\int }_0^x{(x-u)}^{\mu -1}{u}^{\beta -1}{E}_{\alpha ,\beta }[-\eta u^{\alpha }]du=x^{\beta +\mu -1}{E}_{\alpha ,\beta +\mu }[-\eta x^{\alpha }],$$

(127)

where the parameters satisfy the conditions

$$\mu >0,\alpha >0,\beta >0.$$

(128)

Equation (127) can be written in the form

$${\int }_0^x\frac{{\left(\lambda (x-u)\right)}^{\mu -1}}{\Gamma \left(\mu \right)}{\left(\lambda u\right)}^{\beta -1}{E}_{\alpha ,\beta }[-{\left(\lambda u\right)}^{\alpha }]du={\lambda }^{-1}{\left(\lambda x\right)}^{\beta +\mu -1}{E}_{\alpha ,\beta +\mu }[-{\left(\lambda x\right)}^{\alpha }].$$

(129)

Using functions (107) and (110), Equation (129) can be written in the form

$$(M_a\ast M_c)\left(x\right)=(h_{\mu }\left(\lambda x\right)\ast e_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-1}{e}_{\alpha ,\beta +\mu }\left(\lambda x\right).$$

(130)

□

Proposition 7.

Equation

$$(h_{\mu }\left(\lambda x\right)\ast {\omega }_{\alpha }\left(\lambda x\right))={\lambda }^{-1}{\omega }_{\alpha +\mu }\left(\lambda x\right)$$

(131)

holds for all $x>0$, where $\alpha >0$, $\mu >0$, $\lambda >0$.

Proof. 

For function (112), one can use the following example (see Equation (19) in Table 9.1 in [1], p. 172) in the form

$$\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^x{(x-u)}^{\alpha -1}{u}^{\mu /2}{J}_{\mu }\left(\lambda \sqrt{u}\right)du={\left(\frac{2}{\lambda }\right)}^{\alpha }{x}^{(\alpha +\mu )/2}{J}_{\alpha +\mu }\left(\lambda \sqrt{x}\right),$$

(132)

where the parameters satisfy the conditions $\alpha >0$, and $\mu >-1$. Equation (132) can be written ($\mu \to \mu -1$, $\lambda \to 2\sqrt{\lambda }$) in the form

$${\int }_0^x\frac{{\left(\lambda (x-u)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{\left(\sqrt{\lambda u}\right)}^{\mu -1}{J}_{\mu -1}\left(2\sqrt{\lambda u}\right)du={\lambda }^{-1}{\left(\sqrt{\lambda x}\right)}^{\alpha +\mu -1}{J}_{\alpha +\mu -1}\left(2\sqrt{\lambda x}\right),$$

(133)

Using functions (112) and (113), Equation (133) can be rewritten in the form

$$(M_a\ast M_d)\left(x\right)=(h_{\mu }\left(\lambda x\right)\ast {\omega }_{\alpha }\left(\lambda x\right))={\lambda }^{-1}{\omega }_{\alpha +\mu }\left(\lambda x\right).$$

(134)

□

Proposition 8.

Equation

$$(h_{\alpha }\left(\lambda x\right)\ast e^{-\lambda x})={\lambda }^{-1}{e}_{1,\alpha +1}\left(\lambda x\right)$$

(135)

holds for all $x>0$, where $\alpha >0$, $\lambda >0$.

Proof. 

Equation (8) in Table 9.1 in [1], p. 173, has the form

$${\int }_0^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{e}^{-\lambda u}du=x^{\alpha }{E}_{1,\alpha +1}[-\lambda x],$$

(136)

where $\alpha >0$ and $x>0$. Using functions (107) and (110), Equation (136) can be written as

$$(h_{\alpha }\left(\lambda x\right)\ast e^{-\lambda x})={\lambda }^{-1}{e}_{1,\alpha +1}\left(\lambda x\right).$$

(137)

Equation (137) can be considered a special case of Equation (130) for $\alpha =\beta =1$, $\mu =\alpha$, where $E_{1,1}\left[x\right]=exp\left(x\right)$. □

Proposition 9.

Equation

$$(h_{\alpha }\left(\lambda x\right)\ast h_{\beta ,\lambda }\left(\lambda x\right))={\lambda }^{-1}{\varphi }_{\beta ,\alpha +\beta }\left(\lambda x\right)$$

(138)

holds for all $x>0$, where $\alpha >0$, $\beta >0$, $\mu >0$, $\lambda >0$.

Proof. 

Equation (9) in Table 9.1 in [1], p. 173, has the form

$${\int }_0^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}{u}^{\beta -1}{e}^{-\lambda u}du=\frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}{x}^{\alpha +\beta -1}{}_1{F}_1(\beta ;\alpha +\beta ;-\lambda x),$$

(139)

where $\alpha >0$, $\beta >0$, $x>0$, and ${}_1{F}_1(a;b;z)=\Phi (a;b;z)$ is the confluent hypergeometric Kummer function [4], p. 29. Using functions (107), (108), and (116), Equation (139) can be written in the form

$$(M_a\ast M_b)\left(x\right)=(h_{\alpha }\left(\lambda x\right)\ast h_{\beta ,\lambda }\left(\lambda x\right))={\lambda }^{-1}{\varphi }_{\beta ,\alpha +\beta }\left(\lambda x\right)$$

(140)

where $\lambda >0$. □

Proposition 10.

Equation

$$(h_{\mu }\left(\lambda x\right)\ast {\varphi }_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-1}{\varphi }_{\alpha ,\beta +\mu }\left(\lambda x\right)$$

(141)

holds for all $x>0$, where $\alpha >0$, $\beta >0$, $\mu >0$, $\lambda >0$.

Proof. 

Using Equation (22) in Table 9.1 in [1], p. 173, in the form

$${\int }_0^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}\frac{u^{\beta -1}}{\Gamma \left(\beta \right)}{}_2{F}_1(\mu ,\nu ;\beta ;bu)du=\frac{u^{\alpha +\beta -1}}{\Gamma (\alpha +\beta )}{}_2{F}_1(\mu ,\nu ;\alpha +\beta ;bu)$$

(142)

holds for $x>0$, where $\alpha >0$, $\beta >0$, $\mu >0$, $\nu >0$.

Using the parameter $b=-\lambda /\nu$ and the limit

$$\underset{\nu \to \infty }{lim}{}_2{F}_1(\mu ,\nu ;\beta ;-(\lambda /\nu )u)={}_1{F}_1(\mu ;\beta ;-\lambda u),$$

(143)

one can obtain

$${\int }_0^x\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}\frac{u^{\beta -1}}{\Gamma \left(\beta \right)}\Phi (\mu ;\beta ;-\lambda u)du=\frac{u^{\alpha +\beta -1}}{\Gamma (\alpha +\beta )}\Phi (\mu ;\alpha +\beta ;-\lambda u).$$

(144)

As a result, Equation (144) can be written as

$$(M_a\ast M_f)\left(x\right)=(h_{\mu }\left(\lambda x\right)\ast {\varphi }_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-1}{\varphi }_{\alpha ,\beta +\mu }\left(\lambda x\right)$$

(145)

holds for all $x>0$, where $\alpha >0$, $\beta >0$, $\mu >0$, $\lambda >0$. □

5.5. List of Functions and Equations to Derive Multi-Kernels

To build multi-kernel GF operators, the following functions can be considered. For GFI kernels, the parameter values of these functions should be restricted. In order for these functions to describe the operator kernels belonging to the set ${\mathcal{L}}_1$, the parameter values in these functions must be restricted. For example, the parameters of functions (146) and (149) are $\alpha \in (0,1)$; the parameters of functions (148) and (150) are $0<\alpha \le \beta <1$; the parameters of function (147) are $\alpha \in (0,1)$ and $\beta >0$. However, when considering sets ${\mathcal{L}}_n$, ${\mathcal{L}}_n^{<m>}$, and ${\mathcal{L}}_n^{<1|m>}$, the restrictions on the range of values of these parameters change.

$$h_{\alpha }\left(x\right):=\frac{x^{\alpha -1}}{\Gamma \left(\alpha \right)}.$$

(146)

$$h_{\alpha ,\beta }\left(x\right):=\frac{x^{\alpha -1}}{\Gamma \left(\alpha \right)}{e}^{-\beta x}.$$

(147)

$$e_{\alpha ,\beta }\left(x\right):=x^{\beta -1}{E}_{\alpha ,\beta }[-x^{\alpha }].$$

(148)

$${\omega }_{\alpha }\left(x\right):={\left(\sqrt{x}\right)}^{\alpha -1}{J}_{\alpha -1}\left(2\sqrt{x}\right).$$

(149)

$${\varphi }_{\alpha ,\beta }\left(x\right):=x^{\beta -1}\Phi (\alpha ,\beta ;-x).$$

(150)

Note that functions (146), (147), and (149) belong to the set $C_{-1}(0,\infty )$, if $\alpha >0$. Functions (148) and (150) belong to the set $C_{-1}(0,\infty )$, if $\beta >0$.

Let us define the following notation for the Laplace convolution

$$(f\left(\lambda x\right)\ast g\left(\lambda x\right))={\int }_0^{x}f\left(\lambda (x-u)\right)g\left(\lambda u\right)du.$$

(151)

To derive the kernels $M^{<1|m>}$ of the multi-kernel GF integrals and derivatives, the following equations can be used. In these equations, the parameters of the functions are assumed to be positive. Because of this, these equations can be reused any finite number of times for operator kernels belonging to the set ${\mathcal{L}}_1$ to obtain the kernels $M^{<1|m>}$.

$$(h_{\alpha }\left(\lambda x\right)\ast h_{\beta }\left(\lambda x\right))={\lambda }^{-1}{h}_{\alpha +\beta }\left(\lambda x\right).$$

(152)

$$(h_{\alpha ,\lambda }\left(\lambda x\right)\ast h_{\beta ,\lambda }\left(\lambda x\right))={\lambda }^{-1}{h}_{\alpha +\beta ,\lambda }\left(\lambda x\right).$$

(153)

$$(h_{\mu }\left(\lambda x\right)\ast e_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-1}{e}_{\alpha ,\beta +\mu }\left(\lambda x\right).$$

(154)

As a special case of this equation, one can consider

$$(h_{\alpha }\left(\lambda x\right)\ast e^{-\lambda x})={\lambda }^{-1}{e}_{1,\alpha +1}\left(\lambda x\right)$$

(155)

where $e_{1,1}\left(x\right)=exp(-x)$. It should be noted that the function $e_{\alpha ,\beta }\left(\lambda x\right)$ belonging to the set ${\mathcal{L}}_1$ is $0<\alpha \le \beta <1$. Therefore, $e_{1,1}\left(x\right)=exp(-x)$ cannot belong to the set ${\mathcal{L}}_1$.

$$(h_{\mu }\left(\lambda x\right)\ast {\omega }_{\alpha }\left(\lambda x\right))={\lambda }^{-1}{\omega }_{\alpha +\mu }\left(\lambda x\right).$$

(156)

$$(h_{\alpha }\left(\lambda x\right)\ast h_{\beta ,\lambda }\left(\lambda x\right))={\lambda }^{-1}{\varphi }_{\beta ;\alpha +\beta }\left(\lambda x\right).$$

(157)

$$(h_{\mu }\left(\lambda x\right)\ast {\varphi }_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-1}{\varphi }_{\alpha ,\beta +\mu }\left(\lambda x\right).$$

(158)

In these equations, one can consider $x>0$, and the parameters $\alpha >0$, $\beta >0$, and $\lambda >0$.

5.6. Examples of Kernels of Multi-Kernel GF Integrals

In this subsection, some kernels of multi-kernel GF integrals are derived. Using Remark 5, one can state that examples of GFI kernels can be considered examples of GFD kernels, since one can use the kernel pairs $(M_{j,new}={\lambda }^{-1}{K}_j\left(x\right),K_{j,new}=\lambda M_j\left(x\right))$ for each pair $(M_j\left(x\right),K_j\left(x\right))$ (for details, see [48]).

Let us consider the PL-type kernels in the form

$$M_{a|j}\left(x\right)=h_{{\alpha }_j}\left(\lambda x\right),K_{a|j}\left(x\right)=\lambda h_{1-{\alpha }_j}\left(\lambda x\right),$$

(159)

where all ${\alpha }_j\in (0,1)$, $\lambda >0$ and $j=1,\dots ,m$. Kernel pair (159) belongs to the Luchko set ${\mathcal{L}}_1$ for all $j=1,\dots ,m$, where $m\in \mathbb{N}$. Then, the kernel pair

$$(h_{n-1}\ast M_{a|j})\left(x\right)=(h_{n-1}\ast h_{{\alpha }_j})\left(x\right)=h_{{\alpha }_j+n-1}\left(x\right),K_{a|j}\left(x\right)=h_{1-{\alpha }_j}\left(x\right),$$

(160)

belongs to the Luchko set ${\mathcal{L}}_n$ for all $j=1,\dots ,m$, where $n,m\in \mathbb{N}$.

In the multi-kernel case, one can consider the kernels

$$M^{<1|m>}\left(x\right)=(h_{m(n-1)}\ast M_{a|1}\ast \dots \ast M_{a|m})\left(x\right)=(h_{m(n-1)}\ast h_{\alpha })\left(x\right)=h_{\alpha +m(n-1)}\left(x\right),$$

(161)

$$K^{<1|m>}\left(x\right)=(K_{a|1}\ast \dots \ast K_{a|m})\left(x\right)=h_{m-\alpha }\left(x\right),$$

(162)

where

$$\alpha =\sum _{j=1}^m{\alpha }_j,$$

(163)

and ${\alpha }_j\in (0,1)$ for all $j=1,\dots ,m$. The pair of kernels (161) and (162) belongs to the set ${\mathcal{L}}_n^{<1|m>}$, where $n,m\in \mathbb{N}$.

As a result, the power-law functions of the form $h_{\alpha }\left(x\right)$ can be used as kernels (161) and (162) of the multi-kernel GFI and GFD for all $\alpha >0$ instead of $\alpha \in (0,1)$.

Note that

$$(M^{<1|m>}\ast K^{<1|m>})\left(x\right)=(h_{\alpha +m(n-1)}\ast h_{m-\alpha })\left(x\right)=h_{mn}\left(x\right)={\left\{1\right\}}^{mn}\left(x\right),$$

(164)

where $0<\alpha <m$ since $0<{\alpha }_j<1$ for all $j=1,\dots ,m$.

Similarly, one can derive different operator kernels for the multi-kernel GF integrals and derivatives of arbitrary orders. Let us give examples of the kernels $M^{<1|m>}\left(x\right)$.

Proposition 11.

Let $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ such that $M_j\left(x\right)=h_{{\alpha }_j}\left(\lambda x\right)$ for all $j=1,\dots ,m$, where ${\alpha }_j\in (0,1)$.

Then, the kernel of the multi-kernel GFI has the form

$$M_a^{<1|m>}\left(x\right)={\lambda }^{-mn+1}{h}_{\alpha +m(n-1)}\left(\lambda x\right),$$

(165)

where

$$\alpha =\sum _{j=1}^m{\alpha }_j,$$

(166)

and ${\alpha }_j\in (0,1)$ for all $j=1,\dots ,m$.

Proof. 

Using Equation (152) (see also (122)), one can obtain

$$(h_{{\alpha }_1}\left(\lambda x\right)\ast \dots \ast h_{{\alpha }_m}\left(\lambda x\right))={\lambda }^{-(m-1)}{h}_{\alpha }\left(\lambda x\right),$$

(167)

where $\alpha$ is defined by Equation (166), and

$$M_a^{<1|m>}\left(x\right)=(h_{m(n-1)}\left(x\right)\ast {\lambda }^{-m+1}{h}_{\alpha }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-m+1}(h_{m(n-1)}\left(\lambda x\right)\ast h_{\alpha }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-m+1}{\lambda }^{-1}{h}_{\alpha +m(n-1)}\left(\lambda x\right)={\lambda }^{-mn+1}{h}_{\alpha +m(n-1)}\left(\lambda x\right),$$

(168)

where $\alpha$ is defined by Equation (166). Note that $\left[M_a^{<1|m>}\left(x\right)\right]={\left[x\right]}^{mn-1}$. □

Proposition 12.

Let $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ such that $M_j\left(x\right)=h_{{\alpha }_j,\lambda }\left(\lambda x\right)$ for all $j=1,\dots ,m$, where ${\alpha }_j\in (0,1)$.

Then, the kernel of the multi-kernel GFI has the form

$$M_b^{<1|m>}\left(x\right)={\lambda }^{-mn+1}{\varphi }_{\alpha ,\alpha +m(n-1)}\left(\lambda x\right),$$

(169)

where

$$\alpha =\sum _{j=1}^m{\alpha }_j.$$

(170)

and ${\alpha }_j\in (0,1)$ for all $j=1,\dots ,m$.

Proof. 

Using Equation (153) (see also Equations (125) and (140)), one can obtain

$$(h_{{\alpha }_1,\lambda }\left(\lambda x\right)\ast \dots \ast h_{{\alpha }_m,\lambda }\left(\lambda x\right))={\lambda }^{-(m-1)}{h}_{\alpha ,\lambda }\left(\lambda x\right),$$

(171)

where $\alpha$ is defined by Equation (170), and

$$M_b^{<1|m>}\left(x\right)=(h_{m(n-1)}\left(x\right)\ast {\lambda }^{-m+1}{h}_{\alpha ,\lambda }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-m+1}(h_{m(n-1)}\left(\lambda x\right)\ast h_{\alpha ,\lambda }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-m+1}{\lambda }^{-1}{\varphi }_{\alpha ,\alpha +m(n-1)}\left(\lambda x\right)={\lambda }^{-mn+1}{\varphi }_{\alpha ,\alpha +m(n-1)}\left(\lambda x\right).$$

(172)

□

Proposition 13.

Let $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ such that $M_j\left(x\right)=h_{{\alpha }_j}\left(\lambda x\right)$ for all $j=1,\dots ,(m-1)$, where ${\alpha }_j\in (0,1)$, and let $M_m\left(x\right)=e_{\alpha ,\beta }\left(\lambda x\right)$ for $j=m$, where $0<\alpha \le \beta <1$.

Then, the kernel of the multi-kernel GFI has the form

$$M_c^{<1|m>}\left(x\right)={\lambda }^{-(mn-1)}{e}_{\alpha ,\beta +\mu +m(n-1)}\left(\lambda x\right),$$

(173)

where $0<\alpha \le \beta <1$, and

$$\mu =\sum _{j=1}^{m-1}{\alpha }_j,$$

(174)

where ${\alpha }_j\in (0,1)$ for all $j=1,\dots ,(m-1)$.

Proof. 

Using the equality

$$({\lambda }^{-m-2}{h}_{\mu }\left(\lambda x\right)\ast e_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-(m-1)}{e}_{\alpha ,\beta +\mu }\left(\lambda x\right),$$

(175)

where $\mu$ is defined by Equation (174), one can obtain

$$M_c^{<1|m>}\left(x\right)=(h_{m(n-1)}\left(x\right)\ast {\lambda }^{-(m-1)}{e}_{\alpha ,\beta +\mu }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-(m-1)}(h_{m(n-1)}\left(\lambda x\right)\ast e_{\alpha ,\beta +\mu }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-(m-1)}{\lambda }^{-1}{e}_{\alpha ,\beta +\mu +m(n-1)}\left(\lambda x\right)={\lambda }^{-(mn-1)}{e}_{\alpha ,\beta +\mu +m(n-1)}\left(\lambda x\right),$$

(176)

where $0<\alpha \le \beta <1$, $\lambda >0$. □

Proposition 14.

Let $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ such that $M_j\left(x\right)=h_{{\alpha }_j}\left(\lambda x\right)$ for all $j=1,\dots ,(m-1)$, where ${\alpha }_j\in (0,1)$, and let $M_m\left(x\right)={\omega }_{\alpha }\left(\lambda x\right)$ for $j=m$, where $0<\alpha <1$.

Then, the kernel of the multi-kernel GFI has the form

$$M_d^{<1|m>}\left(x\right)={\lambda }^{-(mn-1)}{\omega }_{\alpha +\mu +m(n-1)}\left(\lambda x\right).$$

(177)

where $0<\alpha <1$ and

$$\mu =\sum _{j=1}^{m-1}{\alpha }_j,$$

(178)

where ${\alpha }_j\in (0,1)$ for all $j=1,\dots ,(m-1)$.

Proof. 

Using the equality

$$(h_{{\alpha }_1}\left(\lambda x\right)\ast \dots \ast h_{{\alpha }_{m-1}}\left(\lambda x\right)={\lambda }^{-(m-2)}{h}_{\mu }\left(\lambda x\right),$$

(179)

where $\mu$ is defined by Equation (178), and

$$({\lambda }^{-(m-2)}{h}_{\mu }\left(\lambda x\right)\ast {\omega }_{\alpha }\left(\lambda x\right))={\lambda }^{-(m-1)}{\omega }_{\alpha +\mu }\left(\lambda x\right),$$

(180)

one can obtain

$$M_d^{<1|m>}\left(x\right)=(h_{m(n-1)}\left(x\right)\ast {\lambda }^{-(m-1)}{\omega }_{\alpha +\mu }\left(\lambda x\right))={\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-(m-1)}(h_{m(n-1)}\left(\lambda x\right)\ast {\omega }_{\alpha +\mu }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-(m-1)}{\lambda }^{-1}{\omega }_{\alpha +\mu +m(n-1)}\left(\lambda x\right)={\lambda }^{-(mn-1)}{\omega }_{\alpha +\mu +m(n-1)}\left(\lambda x\right),$$

(181)

where $0<\alpha <1$ and $\lambda >0$. □

Proposition 15.

Let $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ such that $M_j\left(x\right)=h_{{\alpha }_j}\left(\lambda x\right)$ for all $j=1,\dots ,k$, and let $M_j\left(x\right)=h_{{\alpha }_j,\lambda }\left(\lambda x\right)$ for all $j=k+1,\dots ,m)$, where ${\beta }_{j}in(0,1)$ for all $j=1,\dots ,m$.

Then, the kernel of the multi-kernel GFI has the form

$$M_f^{<1|m>}\left(x\right)={\lambda }^{-(mn-1)}{\varphi }_{\nu ,m(n-1)+\eta +\nu }\left(\lambda x\right).$$

(182)

where

$$\eta =\sum _{j=1}^k{\alpha }_j,\nu =\sum _{j=k+1}^m{\alpha }_j.$$

(183)

Proof. 

Using parameters (183), one can obtain the following

$$(h_{{\alpha }_1}\left(\lambda x\right)\ast \dots \ast h_{{\alpha }_k}\left(\lambda x\right))={\lambda }^{-(k-1)}{h}_{\eta }\left(\lambda x\right),$$

(184)

$$(h_{{\alpha }_{k+1},\lambda }\left(\lambda x\right)\ast \dots \ast h_{{\alpha }_m,\lambda }\left(\lambda x\right))={\lambda }^{-(m-k-1)}{h}_{\nu ,\lambda }\left(\lambda x\right),$$

(185)

where $\mu$ and $\nu$ are defined by Equation (183). Then,

$$M_f^{<1|m>}\left(x\right)=(h_{m(n-1)}\left(x\right)\ast {\lambda }^{-(k-1)}{h}_{\eta }\left(\lambda x\right)\ast {\lambda }^{-(m-k-1)}{h}_{\nu ,\lambda }\left(\lambda x\right))=$$

$${\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-(k-1)}{\lambda }^{-(m-k-1)}(h_{m(n-1)}\left(\lambda x\right)\ast h_{\eta }\left(\lambda x\right)\ast h_{\nu ,\lambda }\left(\lambda x\right))=$$

$${\lambda }^{-(mn-3)}{\lambda }^{-1}(h_{m(n-1)+\eta }\left(\lambda x\right)\ast h_{\nu ,\lambda }\left(\lambda x\right))=$$

$${\lambda }^{-(mn-2)}{\varphi }_{\nu ,m(n-1)+\eta +\nu }\left(\lambda x\right)={\lambda }^{-(mn-1)}{\varphi }_{\nu ,m(n-1)+\eta +\nu }\left(\lambda x\right).$$

(186)

□

Proposition 16.

Let $(M_j,K_j)$ belong to the Luchko set ${\mathcal{L}}_1$ such that $M_j\left(x\right)=h_{{\alpha }_j}\left(\lambda x\right)$ for all $j=1,\dots ,(m-1)$, where ${\alpha }_j\in (0,1)$, and let $M_m\left(x\right)={\varphi }_{\alpha ,\beta }\left(\lambda x\right)$ for $j=m$, where $0<\alpha \le \beta <1$.

Then, the kernel of the multi-kernel GFI has the form

$$M_g^{<1|m>}\left(x\right)={\lambda }^{-(mn-1)}{\varphi }_{\alpha ,\beta +\mu +m(n-1)}\left(\lambda x\right),$$

(187)

where $0<\alpha \le \beta <1$, and

$$\mu =\sum _{j=1}^{m-1}{\alpha }_j.$$

(188)

Proof. 

Using the equalities

$$(h_{{\alpha }_1}\left(\lambda x\right)\ast \dots \ast h_{{\alpha }_{m-1}}\left(\lambda x\right))={\lambda }^{-(m-2)}{h}_{\mu }\left(\lambda x\right),$$

(189)

and

$$(h_{m(n-1)}\left(x\right)\ast {\lambda }^{-(m-2)}{h}_{\mu }\left(\lambda x\right))={\lambda }^{-\left(m\right(n-1)-1)}{\lambda }^{-(m-2)}(h_{m(n-1)}\left(\lambda x\right)\ast h_{\mu }\left(\lambda x\right))=$$

$${\lambda }^{-(mn-3)}{\lambda }^{-1}{h}_{m(n-1)+\mu }\left(\lambda x\right)$$

(190)

where $\mu$ is defined by Equation (188), one can obtain

$$M_g^{<1|m>}\left(x\right)={\lambda }^{-(mn-2)}(h_{m(n-1)+\mu }\left(\lambda x\right)\ast {\varphi }_{\alpha ,\beta }\left(\lambda x\right))={\lambda }^{-(mn-1)}{\varphi }_{\alpha ,\beta +\mu +m(n-1)}\left(\lambda x\right).$$

(191)

□

6. Conclusions

Let us briefly list the main results obtained in this article. The multi-kernel GFC of an arbitrary order on the interval $(0,\infty )$ is proposed as a generalization of Luchko’s single-kernel GFC of an arbitrary order, described in [27,29,30,31], and the multi-kernel approach suggested in [35]. The multi-kernel GFC is also proposed for the finite intervals $(a,b)$, where $-\infty <a<b<\infty$. These calculi can be considered one of the possible solutions to Problem 1 of general fractional calculus, described by Luchko in Section “2.4 Open problems” of his work [37], p. 3255.