Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives

Abstract

In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D’Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed.

1. Introduction and Background: Prabhakar Function in Operator Kernels

In fractional calculus, various types of fractional integrals and derivatives are known (for example, see books [1,2,3,4,5] and handbooks [6,7]). Fractional calculus (FC) allows us to describe different types of effects and phenomena in physical, biological, social, economic, and other sciences. For example, it is applied in viscoelasticity [8], fractional dynamics [9,10,11], physical kinetics [12], mechanics of continuum [13], and almost all areas of physics [14,15], biology [16] and economics [17,18]. One can distinguish the following types of phenomena by some properties of kernels of fractional operators: power-law frequency dispersion; fading memory (forgetting); power-law spatial dispersion; spatial nonlocality; distributed lag; and distributed scaling (dilation). These effects and phenomena are characterized by special types of operator kernels [19]. One can impose mathematical conditions on the kernels [19], which allow us to uniquely identify various types of phenomena.

In economics, an important condition that is imposed on the operator kernel is the equality $K\left(0\right)=1$ or $K\left(t,t\right)=1$ to describe the depreciation of equipment, depreciation of fixed assets (of capital), wear, tear, obsolescence, and aging [20] (see also [21,22]). In economic models, exponential functions and the probability density function of the exponential distribution are often used to describe depreciation.

This article discusses a new type of phenomenon, for which conditions on operator kernels can be written exactly. The possibility of using some fractional derivatives and integrals to describe depreciation and obsolescence phenomena is analyzed. In general, the capital fixed assets that existed at time $t=a$ will gradually decrease due to depreciation—both material and moral. In this case, the operator kernel can be interpreted as a function that characterizes depreciation (for example, the depreciation of the capital fixed assets). Depreciation is the process of transferring the value of fixed assets to the value of manufactured and sold final products as they wear out—both material and moral. Note that the main causes of depreciation are obsolescence and natural wear and tear.

It is usually assumed that the operator kernel (depreciation function) is a certain decreasing function, which is assumed to be equal to unity at zero. In other words, the operator kernel describing depreciation should not be singular at zero [20,21,22]. Usually an exponential function is used as a function of depreciation. However, such a kernel does not allow simultaneous consideration of memory effects and non-exponential depreciation effects.

As a tool to describe depreciation of a non-exponential type, while simultaneously taking into account memory effects, we propose to use the Prabhakar function [23,24,25,26,27] as the kernel of operators. Fractional calculus with operators that contain the Prabhakar (generalized Mittag-Leffler) function in the kernel, along with its application, is described in various papers (for example, see works on this fractional calculus [26,27,28,29,30,31,32,33] and its applications [34,35,36,37,38,39,40]). In this paper, we use some of the results of these works. Important restrictions on the formulation of the fractional calculus with fractional derivatives and integrals with nonsingular kernels (the Mittag-Leffler functions) should also be emphasized (for details, see [41,42]). Note that a special case of the Prabhakar function is used in the general fractional calculus (for example, see [43] (pp. 18–19), and [44] (p. 593)). Note that applications can use operators that are sequential operations of two or more fractional derivatives and integrals of different types [19,45].

1.1. Prabhakar Function

In 1971, Tilak R. Prabhakar [23] introduced the entire function

$$E_{\rho ,\mu }^{\gamma }\left[z\right]={\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{\Gamma \left(\rho k+\mu \right)} \frac{z^k}{k!},$$

(1)

where $\rho ,\mu ,\gamma \in ℂ$ and $Re\left(\rho \right)>0,$ and ${\left(\gamma \right)}_k$ is the Pochhammer symbol that is defined by the following equations:

$${\left(\gamma \right)}_k=\gamma \left(\gamma +1\right)\left(\gamma +2\right)\dots \left(\gamma +n-1\right)=\frac{\Gamma \left(\gamma +k\right)}{\Gamma \left(\gamma \right)},$$

(2)

$${\left(-\gamma \right)}_k={\left(-1\right)}^k{\left(\gamma -k+1\right)}_k={\left(-1\right)}^k\frac{\Gamma \left(\gamma +1\right)}{\Gamma \left(\gamma -k+1\right)}.$$

(3)

For negative integer values $\gamma =-n$, $n\in \mathbb{N},$ the Prabhakar function is the $n$-degree polynomial

$$E_{\rho ,\mu }^{-n}\left[z\right]={\displaystyle \sum }_{k=0}^n{\left(-1\right)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\frac{z^k}{\Gamma \left(\rho k+\mu \right)} ,$$

(4)

where $\left(\begin{array}{c}n\\ k\end{array}\right)$ are the binomial coefficients [1,4,25,27]. In the special case $\gamma =-1$, the Prabhakar function is represented in the following form:

$$E_{\rho ,\mu }^{-1}\left[z\right]={\displaystyle \sum }_{k=0}^1{\left(-1\right)}^k\left(\begin{array}{c}1\\ k\end{array}\right)\frac{z^k}{\Gamma \left(\rho k+\mu \right)}=\frac{1}{\Gamma \left(\mu \right)}-\frac{z}{\Gamma \left(\rho +\mu \right)}.$$

(5)

The Prabhakar function $E_{\rho ,\mu }^{\gamma }\left[z\right]$ is also called the three-parameter Mittag-Leffler function [25]. We should note that Prabhakar proposed [23] integral operators with Function (1) in the kernel. Today, these operators are usually called the Prabhakar fractional integrals.

The Prabhakar fractional integral was proposed in 1971 [23]. The left-inverse operators—which are usually called fractional derivatives—for the Prabhakar fractional integrals were first proposed in 2004 in the work of Anatoly A. Kilbas, Megumi Saigo, and Ram K. Saxena [27]. These fractional derivatives contain the Prabhakar functions in the kernel.

Let us note the specificity of the operator kernels that contain the Prabhakar functions, in comparison with the kernels of some other fractional operators.

1.2. Operator Kernels and Properties of the Prabhakar Function as a Kernel

The kernel of the Riemann–Liouville fractional integral is

$$K_{RLI}\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{t}^{\alpha -1},$$

(6)

where $\alpha >0$ [4] (p. 69). It is clear that

$$K_{RLI}\left(0+\right)=\{\begin{array}{c}0\\ 1\\ +\infty \end{array}\begin{array}{c}if\\ if\\ if\end{array} \begin{array}{c}\alpha >1\\ \alpha =1\\ 0<\alpha <1\end{array}.$$

(7)

Therefore, Kernel (6) of the Riemann–Liouville fractional integral can demonstrate three types of behavior at zero ($t=0+$), but the behavior $K_{PI}\left(0\right)=\mathrm{const}$ cannot be realized for non-integer orders $\alpha >0$.

The kernel of the Riemann–Liouville and Caputo fractional derivatives is

$$K_{CD}\left(t\right)=K_{RLD}\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{t}^{n-\alpha -1},$$

(8)

where $n=\left[\alpha \right]+1$, and $n-1<\alpha <n$ for non-integer values of order $\alpha$ [4] (pp. 70–91). It is clear that

$$K_{CD}\left(0+\right)=\{\begin{array}{c}0\\ 1\\ +\infty \end{array}\begin{array}{c}if\\ if\\ if\end{array} \begin{array}{c}0<\alpha <n-1\\ \alpha =n-1\\ \alpha >n-1\end{array}.$$

(9)

This means that the kernel of the Caputo and Riemann–Liouville fractional derivatives can demonstrate only one (singular) type of behavior at zero ($t=0+$) for non-integer orders. The other two cases ($K_{CD}\left(t\right)=0$ and $K_{CD}\left(0+\right)=1$) are not implemented for the following reasons: (A) The case $\alpha =n-1$ cannot be used for the Caputo derivative, since $\alpha =n$ should be used for integer values of $\alpha$ (see equation 2.4.3 in [4] (p. 91)). For this case ($\alpha =n$), the Riemann–Liouville fractional derivative is a standard derivative of integer order. (B) The case $0<\alpha <n-1$ cannot be used by definition of the Caputo and Riemann–Liouville fractional derivatives that contain the condition $n-1<\alpha <n$ for non-integer values of the order $\alpha >0$.

As a result, the power-law kernels of fractional derivatives have significantly less variability in their behavior properties at zero, even in comparison with the fractional integrals to which they are left-inverse. Note that the variety of properties of operator kernels at zero is important for applications of these operators in economics and physics, for example.

The kernel of the Prabhakar fractional integral has the following form:

$$K_{PI}\left(t\right)=t^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega t^{\rho }\right].$$

(10)

Using the definition of the Prabhakar (three parameter Mittag-Leffler) function

$$K_{PI}\left(t\right)=t^{\mu -1}{\displaystyle \sum }_{k=0}^{\infty }\frac{\Gamma \left(\gamma +k\right)}{\Gamma \left(\gamma \right)\Gamma \left(\rho k+\mu \right)} \frac{{\left(\omega t^{\rho }\right)}^k}{k!}=\frac{t^{\mu -1}}{\Gamma \left(\mu \right)}+t^{\mu -1}{\displaystyle \sum }_{k=1}^{\infty }\frac{\Gamma \left(\gamma +k\right)}{\Gamma \left(\gamma \right)\Gamma \left(\rho k+\mu \right)} \frac{{\left(\omega t^{\rho }\right)}^k}{k!},$$

(11)

we see that

$$K_{PI}\left(0+\right)=\{\begin{array}{c}0\\ 1\\ +\infty \end{array}\begin{array}{c}if\\ if\\ if\end{array} \begin{array}{c}\mu >1\\ \mu =1\\ 0<\mu <1\end{array}.$$

(12)

This means that the kernel of the Prabhakar fractional integral can demonstrate three types of behavior at zero.

Let us consider the kernel of the Prabhakar fractional derivative

$$K_{PD}\left(t\right)=t^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega t^{\rho }\right],$$

(13)

where $n\ge \left[Re\left(\mu \right)\right]+1$ with $Re\left(\mu \right)>0$ (for details see below). Using the definition of the Prabhakar (three parameter Mittag-Leffler) function in the form

$$K_{PD}\left(t\right)=t^{n-\mu -1}{\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(-\gamma \right)}_k}{\Gamma \left(\rho k+n-\mu \right)} \frac{{\left(\omega t^{\rho }\right)}^k}{k!}=\frac{t^{n-\mu -1}}{\Gamma \left(n-\mu \right)}+t^{n-\mu -1}{\displaystyle \sum }_{k=1}^{\infty }\frac{{\left(-\gamma \right)}_k}{\Gamma \left(\rho k+n-\mu \right)} \frac{{\left(\omega t^{\rho }\right)}^k}{k!},$$

(14)

one can obtain the following properties of $K_{PD}\left(t\right)$ at the initial point:

$$K_{PD}\left(0+\right)=\{\begin{array}{c}0\\ 1\\ +\infty \end{array}\begin{array}{c}if\\ if\\ if\end{array} \begin{array}{c}0<\mu <n-1\\ \mu =n-1\\ \mu >n-1\end{array}.$$

(15)

This means that the kernel of the Prabhakar fractional derivative can also demonstrate all three types of behavior at zero. Note that this operator remains a fractional operator and under the condition $\mu =n-1$. This behavior significantly distinguishes the Prabhakar operators from other fractional derivatives, which usually have a singularity at zero for kernels $K\left(t,\tau \right)=K\left(t-\tau \right)$, and at $t=\tau$ for kernels $K\left(t,\tau \right)\ne K\left(t-\tau \right)$. It should be emphasized that the kernels of the Prabhakar fractional derivatives, which are proposed in [27], can be used for all positive integer values $n\ge \left[Re\left(\mu \right)\right]+1$, where $Re\left(\mu \right)>0$.

Note that for some processes, an important condition that is imposed on the operator kernel is the equality $K\left(0+\right)=1$ or $K\left(t,t\right)=1$. For example, in economics, this condition is used for the kernels that describe the depreciation of fixed assets (of capital), depreciation of equipment, obsolescence, aging, wear, and tear [20] (see also books [20,21,22]). The kernel $K\left(t,\tau \right)$ or $K\left(t-\tau \right)$ characterizes the share of fixed assets put into operation at time $\tau ,$ and continuing to operate at time $t>\tau$. Obviously, in this case, the condition $K\left(t,t\right)=1$ or $K\left(0+\right)=1$ must be satisfied. For this, economic models often use the exponential functions and the probability density function of the exponential distribution.

To satisfy the initial conditions $K\left(0+\right)=1$ for the operator kernels, one can use the kernels with the Prabhakar function in the forms (10) and (13). These kernels allow us to use the fractional integrals and derivatives that were proposed in [23,27,37,38], with the Prabhakar function in the kernel, to describe depreciation processes in economics.

In addition, one can state that the kernel $K_{PI}\left(t\right)$ is the complete monotonic function for the case $\omega <0$, $0<\rho ,\mu \le 1$, $0<\gamma \le \mu /\rho$. The property of complete monotonicity is important for the interpretation of operator kernels that describe standard depreciation phenomena. However, one can assume that the requirement of complete monotonicity for depreciation kernels is not necessary when taking into account modernization of the equipment.

1.3. Asymptotic Properties of Operator Kernels with Prabhakar Function

Let us consider the asymptotic behavior of the Prabhakar function that is described in [46] (pp. 332–333), and the operator kernels for $\left|z\right|\to \infty$. The asymptotic expansions as $\left|z\right|\to \infty$ are expressed [46] (pp. 332–333) through the following functions:

$$E\left(z\right)=\frac{z^{\raisebox{1ex}{$\left(\gamma -\beta \right)$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}\exp \left(z^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}\right)}{\Gamma \left(\gamma \right)}{\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(-1\right)}^k}{k!}{A}_k{z}^{-\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.},$$

(16)

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

(17)

where $A_k=A_k\left(\alpha ,\beta ,\gamma \right)$ are coefficients. The Prabhakar function asymptotics are

$$E_{\alpha ,\beta }^{\gamma }\left[z\right]~\{\begin{array}{c}E\left(z\right)+H\left(z\exp \left(\mp \pi i\right)\right)\\ H\left(z \exp \left(\mp \pi i\right)\right)\\ E\left(z\right)+E\left(z\exp \left(\mp 2\pi i\right)\right)+H\left(z \exp \left(\mp \pi i\right)\right)\end{array}\begin{array}{c}\left|\arg \left(z\right)\right|\le \inf \left\{\pi -\epsilon ,\alpha \pi -\epsilon \right\}\\ \alpha \pi +\epsilon \le \left|\arg \left(z\right)\right|\le \pi ,\alpha \in \left(0,1\right)\\ \left|\arg \left(z\right)\right|\le \pi ,1<\alpha \le 2\end{array},$$

(18)

where the upper or lower signs are chosen according to $\arg \left(z\right)>0$ or $\arg \left(z\right)<0$, respectively, and

$$E_{\alpha ,\beta }^{\gamma }\left[z\right]~{\displaystyle \sum }_{n=-N}^{N}E\left(z\exp \left(2\pi iN\right)\right),\hspace{1em}\left|\arg \left(z\right)\right|\le \pi ,\hspace{1em}\alpha >2,$$

(19)

where N is the lowest integer satisfying $2N+1>\alpha /2$.

Let us describe the asymptotic behavior (for $\left|z\right|\to \infty$) of the operator kernels that are expressed as a product of $t^{\beta -1}$ and the Prabhakar function $E_{\alpha ,\beta }^{\gamma }\left[z\right]$. In the operator kernels, the real variable $z=\omega t^{\alpha }$, with $t\ge 0$ and $\omega \in ℝ$, is used. In this case, we have the following functions:

$$\begin{gathered} t^{\beta -1}E\left(\omega t^{\alpha }\right)=\frac{{\omega }^{\left(\gamma -\beta \right)/\alpha }{t}^{\gamma -1}\exp \left({\omega }^{1/\alpha }t\right)}{\Gamma \left(\gamma \right)}{\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(-1\right)}^k}{k!}{A}_k{\omega }^{-k/\alpha } t^{-k} \\ =\frac{{\omega }^{\left(\gamma -\beta \right)/\alpha }{\alpha }^{-\gamma }}{\Gamma \left(\gamma \right)}{t}^{\gamma -1}\exp \left({\omega }^{1/\alpha }t\right) \\ +\frac{{\omega }^{\left(\gamma -\beta \right)/\alpha }{t}^{\gamma -1}\exp \left({\omega }^{1/\alpha }t\right)}{\Gamma \left(\gamma \right)}{\displaystyle \sum }_{k=1}^{\infty }\frac{{\left(-1\right)}^k}{k!}{A}_k{\omega }^{-k/\alpha }{t}^{-k}, \end{gathered}$$

(20)

where $A_0={\alpha }^{-\gamma }$, and

$$\begin{gathered} t^{\beta -1}H\left(\omega t^{\alpha }\right)=\frac{t^{\beta -1}{t}^{-\alpha \gamma }}{\Gamma \left(\gamma \right)}{\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(-1\right)}^k}{k!}\frac{\Gamma \left(\gamma +k\right){\omega }^{\gamma -k}}{\Gamma \left(\beta -\alpha \left(\gamma +k\right)\right)}{t}^{-\alpha k} \\ =\frac{{\omega }^{\gamma }}{\Gamma \left(\beta -\alpha \gamma \right)}{t}^{\beta -\alpha \gamma -1}+\frac{t^{\beta -\alpha \gamma -1}}{\Gamma \left(\gamma \right)} {\displaystyle \sum }_{k=1}^{\infty }\frac{{\left(-1\right)}^k}{k!}\frac{\Gamma \left(\gamma +k\right){\omega }^{\gamma -k}}{\Gamma \left(\beta -\alpha \left(\gamma +k\right)\right)}{t}^{-\alpha k}. \end{gathered}$$

(21)

As a result, the following interpretation of the parameters $\alpha$, $\beta$, $\gamma$, and $\omega$ can be formulated. For simplification, the case $0<\alpha <2$ is considered.

Statement 1.

For$\omega >0$, we have the following interpretation of the parameters:

• ($\alpha$) the parameter$\alpha$characterizes the change in the warranted growth rate;
• ($\beta$) the parameter$\beta$characterizes the numerical multipliers only;
• ($\gamma$) the parameter$\gamma$characterizes the power-law change in the amplitude of exponential changes (for example, growth);
• ($\omega$) the parameter$\omega$characterizes the standard growth rate for exponential behavior of the operator kernel.

For$\omega <0$, we have the following interpretation of the parameters: the parameters$\alpha$,$\beta$, and$\gamma$characterize the degrees of power-law change in the combination$\beta -\alpha \gamma$.

There are three types of behavior:$\beta -\alpha \gamma <1$,$\beta -\alpha \gamma =1$, and$\beta -\alpha \gamma >1$.

Using the asymptotic behavior of Functions (20) and (21), it is possible to determine that the asymptotic behavior of the kernels $K_{PI}\left(t\right)$ and $K_{PD}\left(t\right)$ with $\omega <0$ and $\alpha \in \left(0,1\right)$ is described at infinity by the following expressions:

$$\underset{t\to +\infty }{\lim }{K}_{PI}\left(t\right)=\{\begin{array}{c}0\\ {\omega }^{\gamma }/\Gamma \left(\mu -\rho \gamma \right)\\ +\infty \end{array}\begin{array}{c}if\\ if\\ if\end{array} \begin{array}{c}\mu -\rho \gamma <1\\ \mu -\rho \gamma =1\\ \mu -\rho \gamma >1\end{array},$$

(22)

$$\underset{t\to +\infty }{\lim }{K}_{PD}\left(t\right)=\{\begin{array}{c}0\\ \raisebox{1ex}{${\omega }^{-\gamma }$}\!\left/ \!\raisebox{-1ex}{$\Gamma \left(n-\mu +\rho \gamma \right)$}\right.\\ +\infty \end{array}\begin{array}{c}if\\ if\\ if\end{array} \begin{array}{c}n-\mu +\rho \gamma <1\\ n-\mu +\rho \gamma =1\\ n-\mu +\rho \gamma >1\end{array}.$$

(23)

The kernels $K_{PI}\left(t\right)$ and $K_{PD}\left(t\right)$ with $\omega >0$ are described at infinity by the following expressions:

$$\underset{t\to +\infty }{\lim }{K}_{PI}\left(t\right)=+\infty ,\hspace{1em}\underset{t\to +\infty }{\lim }{K}_{PD}\left(t\right)=+\infty ,$$

(24)

where

$$K_{PD}\left(t\right)~\frac{{\omega }^{-\left(\gamma +\beta \right)/\alpha }{\alpha }^{\gamma }}{\Gamma \left(-\gamma \right)}\left(t^{-\gamma -1}+O\left(t^{-\gamma -2}\right)\right)\exp \left({\omega }^{1/\alpha }t\right).$$

(25)

As a result, an exponential growth of the operator kernel is achieved for $t\to +\infty$.

2. Fractional Integrals and Derivatives with the Prabhakar Function in Kernels

2.1. Prabhakar Fractional Integrals and Kilbas–Saigo–Saxena Fractional Derivatives

The Prabhakar integral operator [23,27], which can be called the Prabhakar fractional integral, is defined (see Equation 1.6 in [27] (p. 32)) in the following way:

Definition 1.

The Prabhakar integral operator is defined by the following equation:

$$\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(26)

where $\rho ,\mu ,\gamma ,\omega \in ℂ$, such that$Re\left(\rho \right),Re\left(\mu \right)>0$, and$E_{\rho ,\mu }^{\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]$is the Prabhakar function.

The boundedness of Operator (26) was proven in [27] for the space $L\left(a, b\right)$ of Lebesgue measurable functions on a finite interval $\left[a, b\right],$ and for the space $C\left[a, b\right]$ of continuous functions on $\left[a, b\right]$.

For $\gamma =0$ (and for $\omega =0$) the Prabhakar fractional integral (26) gives the Riemann–Liouville fractional integral of the order $\mu >0$.

Theorem 1 (Kilbas–Saigo–Saxena Theorem).

The operators, which are the left-inversion of the Prabhakar fractional integral, are defined (see Equation 6.5 and Theorem 9 in [27] (p. 47)) in the following form:

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left(D_{RL,a+}^{\mu +\nu }{\epsilon }_{\rho ,\nu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=D_{RL,a+}^{\mu +\nu }\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\nu -1}{E}_{\rho ,\nu }^{-\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(27)

where$\rho ,\mu ,\gamma ,\nu \in ℂ$,$Re\left(\mu \right)>0$,$Re\left(\nu \right)>0$,$Re\left(\rho \right)>0,$and$D_{RL,a+}^{\mu +\nu }$is the Riemann–Liouville derivative of the order$\mu +\nu$. The function$f\left(\tau \right)$belongs to the space$L\left(a, b\right)$of Lebesgue measurable functions on a finite interval$\left[a, b\right]$of the real line.

Proof. 

This theorem is formulated and proved in [27] (p. 47), as in Theorem 9. Equation (27) is Equation 6.5 that is given in [27] (p. 47). □

Remark 1.

The Operator (27) is called the Prabhakar fractional derivative of the Riemann–Liouville type. However, this operator was not proposed in the works of Prabhakar. The first time this operator was proposed was by Anatoly A. Kilbas, Megumi Saigo, and Ram K. Saxena [27] in 2004. Therefore, this operator can be called as the Kilbas–Saigo–Saxena (KSS) fractional derivative. Note that the Kilbas–Saigo–Saxena operator is a left-inverse operator to the Prabhakar fractional integral operator; therefore, this operator is a fractional derivative. As a result, the Kilbas–Saigo–Saxena fractional derivatives and the Prabhakar fractional integrals form a fractional calculus of operators involving a generalized Mittag-Leffler function in the kernels. Therefore, the fractional calculus with the Mittag-Leffler function was proposed in 2004 by Kilbas, Saigo, and Saxena in [27]. Note that any formulation of the fractional calculus for operators with the Mittag-Leffler functions in the kernels has important restrictions, as described in [41,42].

Theorem 2.

The Kilbas–Saigo–Saxena fractional derivative (27) can be represented in the following equivalent form:

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(28)

where$n>\mu$,and$\rho ,\mu ,\gamma ,\omega \in ℂ$, with$Re\left(\mu \right)>0$,$Re\left(\rho \right)>0$.

Proof. 

Using the definition of the Riemann–Liouville fractional integral, one can write the Kilbas–Saigo–Saxena (KSS) fractional derivative as follows:

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left(D_{RL,a+}^{\mu +\nu }{\epsilon }_{\rho ,\nu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\left(I_{RL,a+}^{n-\left(\mu +\nu \right)}{\epsilon }_{\rho ,\nu ,\omega ,a+}^{-\gamma }f\right)\left(t\right),$$

(29)

where $n=\left[Re\left(\mu +\nu \right)\right]+1$ (see Equation 2.1.5 in [4] (p. 70), and Equation 1.11 of [27] (p. 33)). Then, using Theorem 6 of [27] (p. 43), in the form

$$\left(I_{RL,a+}^{\alpha }{\epsilon }_{\rho ,\nu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\nu +\alpha ,\omega ,a+}^{-\gamma }f\right)\left(t\right),$$

(30)

where $\gamma \in ℂ,$ Equation (30) gives

$$\left(I_{RL,a+}^{n-\left(\mu +\nu \right)}{\epsilon }_{\rho ,\nu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,n-\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right),$$

(31)

where $\rho ,\mu ,\gamma ,\omega \in ℂ$, such that $Re\left(\rho \right),Re\left(\mu \right)>0$. Equation (31) allows us to represent the Kilbas–Saigo–Saxena (KSS) fractional derivative in the following form:

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\left({\epsilon }_{\rho ,n-\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau .$$

(32)

It should be emphasized that $n=\left[\mu +\nu \right]+1$. Therefore, one can use the condition $n\ge \left[Re\left(\mu \right)\right]+1$ with $Re\left(\mu \right)>0$, since the parameter $\nu$ is absent in Expression (17). As a result, all positive integer values of $n>\mu$ can be considered. The condition $n=\left[\mu \right]$, which is used in the works [31] (p. 76) and [29] (p. 579), is more restrictive than the condition $n>\mu$ that is actually proposed in [27].

As a result, the Kilbas–Saigo–Saxena fractional derivative can be defined in the following equivalent form:

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(33)

where $n>\mu$, and $\rho ,\mu ,\gamma ,\omega \in ℂ$, $Re\left(\mu \right)>0$, $Re\left(\rho \right)>0$. This form is equivalent to the form that is given in Equation (27).

This ends the proof. □

Remark 2.

Form (33) of the Kilbas–Saigo–Saxena fractional derivative is also given in Definition 6 in [31] (p. 76), and in Equation 17 of [29] (p. 579), but in these works the condition$n=\left[\mu \right]$is used instead of$n\ge \left[Re\left(\mu \right)\right]+1$. In our opinion, the restriction by$n=\left[\mu \right]$is not only unnecessary from a mathematical point of view, but leads to strong restrictions on the use of these operators when describing phenomena such as depreciation and obsolescence. In the case$n\ge \left[Re\left(\mu \right)\right]+1,$the kernel of the Prabhakar fractional derivative can demonstrate three types of behavior at zero ($K_{PD}\left(0\right)$), as described in Equation (15).

Remark 3.

It should be emphasized that the Kilbas–Saigo–Saxena operator (33) is a left-inverse operator to the Prabhakar fractional integral operator. Therefore, the Kilbas–Saigo–Saxena fractional derivative and the Prabhakar fractional integral forms a fractional calculus. Note that$\gamma \in ℂ$. Therefore, one can consider operators with positive and negative values of$\gamma$. For example, the operator

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(34)

where$n>\mu$is also a fractional derivative that is left-inverse to the fractional integral$\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)$. For$\gamma =0$the Kilbas–Saigo–Saxena fractional derivative gives the Riemann–Liouville fractional derivative.

2.2. The D’Ovidio–Polito Operator and Its Modifications

The Prabhakar operator of Caputo type (the regularized Prabhakar fractional derivative) can be also used in the form that was proposed by Mirko D’Ovidio and Federico Polito in [37] in 2013 (see also [38,39] and Equation 19 in [29] (p. 579)). This operator is defined by the following equation:

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,n-\mu ,\omega ,a+}^{-\gamma }{f}^{\left(n\right)}\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f^{\left(n\right)}\left(\tau \right)d\tau ,$$

(35)

where $n>\mu ,$ and $\rho ,\mu ,\gamma ,\omega \in ℂ$, $Re\left(\mu \right)>0$, $Re\left(\rho \right)>0$. Operator (35) can be called the D’Ovidio–Polito (D’OP) fractional derivative. Note that one can use $n>\mu$ instead of $n=\left[\mu \right]$ in the definition of this operator, for the reasons described above.

The Prabhakar fractional derivative of the Hilfer type is proposed in [29]. These operators of the Hilfer type contain the KSS and D’OP fractional derivatives as special cases.

Note that the fractional integrals and derivatives with the Prabhakar functions in the kernels, which were proposed from 1971 to 2014 (for example, see [23,27,37,38]), contain some operators with the Mittag-Leffler function—which have been proposed in recent years—as special cases (for details, see [41,42]).

Remark 4.

Note that the Prabhakar fractional derivative of the Riemann–Liouville type (the KSS fractional derivative) is defined in [27] as a combination of the actions of the Prabhakar fractional integrals and the Riemann–Liouville fractional derivative.

Let us note the following property of the action of the Riemann–Liouville fractional integral on the Prabhakar fractional integral:

$$\left(I_{RL,a+}^{\alpha }{\epsilon }_{\rho ,\nu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu +\alpha ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\nu ,\omega ,a+}^{\gamma }{I}_{RL,a+}^{\alpha }f\right)\left(t\right),$$

(36)

where if $Re\left(\mu \right)>0$, $Re\left(\rho \right)>0,$ and $Re\left(\alpha \right)>0$. For details, see Theorem 6 [27] (p. 43). The action of the Riemann–Liouville fractional derivative on the Prabhakar fractional integral is described by the following expression:

$$\left(D_{RL,a+}^{\alpha }{\epsilon }_{\rho ,\nu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu -\alpha ,\omega ,a+}^{\gamma }f\right)\left(t\right),$$

(37)

if $Re\left(\mu \right)>Re\left(\alpha \right)>0$, $Re\left(\rho \right)>0$, and the special case

$$\left(\frac{d^n}{d{t}^n}{\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu -n,\omega ,a+}^{\gamma }f\right)\left(t\right),$$

(38)

if $Re\left(\mu \right)>n>0$, and $Re\left(\rho \right)>0$. For details, see Theorem 7 in [27] (p. 44).

Remark 5.

Let us consider “new” operators that are combinations of actions of the Prabhakar fractional integrals and the Caputo fractional derivatives in the following form:

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma ,\alpha }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }{D}_{C,a+}^{\alpha }f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]\left(D_{C,a+}^{\alpha }f\right)\left(\tau \right)d\tau .$$

(39)

The special case of this operator is the operator with integer values of$\alpha =n\in \mathbb{N}$,which has the following form:

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma ,n}f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }{f}^{\left(n\right)}f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f^{\left(n\right)}\left(\tau \right)d\tau .$$

(40)

It might seem that we are proposing new fractional operators. In reality, this is not true, since these operators can be expressed through the D’Ovidio–Polito operator (the Prabhakar operator of the Caputo type). Let us prove that Operator (39) is expressed through the D’Ovidio–Polito operator (35). Using$\left(D_{C,a+}^{\alpha }f\right)\left(\tau \right)=\left(I_{RL,a+}^{n-\alpha }{f}^{\left(n\right)}\right)\left(\tau \right)$, one can write this operator as follows:

$$\begin{gathered} \left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma ,\alpha }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }{D}_{C,a+}^{\alpha }f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }{I}_{RL,a+}^{n-\alpha }{f}^{\left(n\right)}\right)\left(t\right) \\ =\left({\epsilon }_{\rho ,\mu +n-\alpha ,\omega ,a+}^{\gamma }{f}^{\left(n\right)}\right)\left(t\right). \end{gathered}$$

(41)

Then, using Equation (35), which defines the D’OP derivative$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\epsilon }_{\rho ,n-\mu ,\omega ,a+}^{-\gamma }{f}^{\left(n\right)}\right)\left(t\right)$, we get

$$\left({\epsilon }_{\rho ,\mu +n-\alpha ,\omega ,a+}^{\gamma }{f}^{\left(n\right)}\right)\left(t\right)=\left({\mathcal{D}}_{\rho ,\alpha -\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right).$$

(42)

if$Re\left(\alpha \right)>Re\left(\mu \right)>0$. It should be noted that$\gamma \in ℂ$. As a result, the proposed Operator (39) can be expressed through the D’Ovidio–Polito operator

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma ,\alpha }f\right)\left(t\right)=\left({\mathcal{D}}_{\rho ,\alpha -\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right).$$

(43)

Equation (43) can also be considered in the special form

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\left({\mathcal{D}}_{\rho ,n-\mu ,\omega ,a+}^{-\gamma ,n}f\right)\left(t\right).$$

(44)

This ends the proof.

Statement 2.

The Kilbas–Saigo–Saxena fractional derivative is the left-inverse operator for the Prabhakar fractional integrals

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }{\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=f\left(t\right),$$

(45)

if$f\left(t\right)\in L\left[a,b\right].$

The proof of this statement is proposed as Theorem 9 in [27] (p. 47).

Statement 3.

The D’Ovidio–Polito operator is a left-inverse operator for the Prabhakar fractional integrals

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }{\epsilon }_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)=f\left(t\right),$$

(46)

if$f\left(t\right)\in C\left[0,b\right].$

The proof of this statement is proposed as Theorem 1 in [33] (pp. 6–7). The proof is based on Equations (38) and (45). Then, the equations

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }{\epsilon }_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)& =\hfill \\ & =\left({\mathit{D}}_{\mathsf{\rho },\mathsf{\mu },\mathsf{\omega },0+}^{\mathsf{\gamma }}{\epsilon }_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)\hfill \\ & -{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\frac{1}{k!}{\left({\epsilon }_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)}^{\left(k\right)}\left(0+\right) \left({\mathit{D}}_{\mathsf{\rho },\mathsf{\mu },\mathsf{\omega },0+}^{\mathsf{\gamma }}{t}^k\right)\left(t\right)\hfill \\ & =f\left(t\right)-{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\frac{1}{k!}{\left({\epsilon }_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)}^{\left(k\right)}\left(0+\right) \left({\mathit{D}}_{\mathsf{\rho },\mathsf{\mu },\mathsf{\omega },0+}^{\mathsf{\gamma }}{t}^k\right)\left(t\right),\hfill \end{array}$$

(47)

and

$${\left({\epsilon }_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)}^{\left(k\right)}\left(0+\right)=\left({\epsilon }_{\rho ,\mu -k,\omega ,0+}^{\gamma }f\right)\left(0+\right)=0,$$

(48)

give (46). This ends the proof.

Remark 6.

Statements 2 and 3 can be called the first fundamental theorems of fractional calculus. The Kilbas–Saigo–Saxena fractional derivative, D’Ovidio–Polito operator, and Prabhakar fractional integral form a fractional calculus. It should be emphasized that the main property of any fractional derivative is to be a left-inverse operator to the corresponding fractional integral. This requirement is important for a self-consistent mathematical theory of the fractional operators to have a fractional calculus of these operators.

Remark 7. 

The D’Ovidio–Polito operator can be interpreted as a combination of fractional derivatives and integrals, namely, the Prabhakar fractional integral, and the Caputo fractional derivatives of integer and non-integer orders. This interpretation is important for applications. In applications, one can have a simultaneous action of two different types of phenomena. For example, in economic processes, one can have the following simultaneous phenomena:

(a) 

the depreciation and distributed time delay;

(b) 

the depreciation and fading memory;

(c) 

the continuously distributed lag and fading memory;

(d) 

the depreciation and scaling;

(e) 

the fading memory and scaling.

From this point of view, one can use the combination of two or more different types of the fractional derivatives and integrals—for example, the operator, which is the combination of the Prabhakar fractional integrals and the Caputo fractional derivatives. As a result, the D’Ovidio–Polito operator can be used as a tool to describe processes with non-exponential depreciation and power-law fading memory.

2.3. Special Cases of the Prabhakar Integral and D’Ovidio–Polito Operator

Let us consider some special cases of the Prabhakar fractional operators.

Example 1.

For$\gamma =1$, we have the equality

$$E_{\rho ,\mu }^1\left[\omega {\left(t-\tau \right)}^{\rho }\right]=E_{\rho ,n}\left[\omega {\left(t-\tau \right)}^{\rho }\right],$$

(49)

and the fractional integral

$$\left({\epsilon }_{\rho ,\mu ,\omega ,a+}f\right)\left(t\right)=\left({\epsilon }_{\rho ,\mu ,\omega ,a+}^{1}f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\mu -1}{E}_{\rho ,\mu }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau .$$

(50)

For$\gamma =-1$,the fractional derivative

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{-1}f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f^{\left(n\right)}\left(\tau \right)d\tau ,$$

(51)

where one can consider$\omega \in ℝ,$and$E_{\alpha ,\beta }\left[z\right]$is the two-parameter Mittag-Leffler function

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

(52)

Example 2.

For$\rho =1$, using the equality

$$E_{1,\mu }^{\gamma }\left[\omega \left(t-\tau \right)\right]=\frac{1}{\Gamma \left(\mu \right)}{F}_{1,1}\left[\gamma ;\mu ;\omega \left(t-\tau \right)\right],$$

(53)

the fractional integral takes the following form:

$$\left({\epsilon }_{\gamma ,\mu ,\omega ,a+}f\right)\left(t\right)=\left({\epsilon }_{1,\mu ,\omega ,a+}^{\gamma }f\right)\left(t\right)=\frac{1}{\Gamma \left(\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\mu -1}{F}_{1,1}\left[\gamma ;\mu ;\omega \left(t-\tau \right)\right]f\left(\tau \right)d\tau .$$

(54)

The fractional derivative can be written as follows:

$$\left({\mathcal{D}}_{1,\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\frac{1}{\Gamma \left(n-\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{F}_{1,1}\left[\gamma ;n-\mu ;\omega \left(t-\tau \right)\right]f^{\left(n\right)}\left(\tau \right)d\tau ,$$

(55)

where$F_{1,1}\left[a;c;z\right]$is the Kummer hypergeometric function

$$F_{1,1}\left[a;c;z\right]={\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(c\right)}_k}\frac{z^k}{k!},$$

(56)

with$a, z\in ℂ,$such that$c\ne 0,-1,-2,\dots$, and Series (56) is absolutely convergent for all$z\in ℂ.$

Example 3.

For$\gamma =-1$, we have the equality

$$E_{\rho ,\mu }^{-1}\left[\omega {\left(t-\tau \right)}^{\rho }\right]=\frac{1}{\Gamma \left(\mu \right)}-\frac{\omega {\left(t-\tau \right)}^{\rho }}{\Gamma \left(\rho +\mu \right)},$$

(57)

and the fractional derivative with$\gamma =1$is represented in the following form:

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{1}f\right)\left(t\right)& =\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{-1}\left[\omega {\left(t-\tau \right)}^{\rho }\right]f^{\left(n\right)}\left(\tau \right)d\tau \hfill \\ & =\frac{1}{\Gamma \left(n-\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{f}^{\left(n\right)}\left(\tau \right)d\tau \hfill \\ & -\frac{\omega }{\Gamma \left(\rho +n-\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n+\rho -\mu -1}{f}^{\left(n\right)}\left(\tau \right)d\tau \hfill \\ & =\left(I_{RL,a+}^{n-\mu }{f}^{\left(n\right)}\right)\left(t\right)-\omega \left(I_{RL,a+}^{\rho +n-\mu }{f}^{\left(n\right)}\right)\left(t\right)\hfill \\ & =\left(D_{C,a+}^{\mu }f\right)\left(t\right)-\omega \left(I_{RL,a+}^{\rho }{D}_{C,a+}^{\mu }f\right)\left(t\right),\hfill \end{array}$$

(58)

where we use the semi-group property of the Riemann–Liouville fractional integral.

As a result, we get

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{1}f\right)\left(t\right)=\left(D_{C,a+}^{\mu }f\right)\left(t\right)-\omega \left(I_{RL,a+}^{\rho }{D}_{C,a+}^{\mu }f\right)\left(t\right).$$

(59)

As a result, the D’Ovidio–Polito operator with$\gamma =1$is represented through the Caputo fractional derivative.

Example 4.

For$\gamma =1,$and$\mu =1$, we have the equality

$$E_{\rho ,1}^1\left[\omega {\left(t-\tau \right)}^{\rho }\right]=E_{\rho }\left[\omega {\left(t-\tau \right)}^{\rho }\right],$$

(60)

and the fractional integral

$$\left({\epsilon }_{\rho ,1,\omega ,a+}^{1}f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{E}_{\rho }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(61)

where$E_{\alpha }\left[z\right]$is the classical (one parameter) Mittag-Leffler function

$$E_{\alpha }\left[z\right]={\displaystyle \sum }_{k=0}^{\infty }\frac{z^k}{\Gamma \left(\rho k+1\right)}.$$

(62)

For$\gamma =-1$and$n-\mu =1$, we have the fractional derivative

$$\left({\mathcal{D}}_{\rho ,n-1,\omega ,a+}^{-1}f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{E}_{\rho }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f^{\left(n\right)}\left(\tau \right)d\tau ,$$

(63)

where one can consider$\omega \in ℝ$.

Example 5.

For$\gamma =1$,$\mu =1$, and$\rho =1$, we have the equality

$$E_{1,1}^1\left[\omega \left(t-\tau \right)\right]=exp\left[\omega \left(t-\tau \right)\right],$$

(64)

and the fractional integral

$$\left({\epsilon }_{1,1,\omega ,a+}^{1}f\right)\left(t\right)={{\displaystyle \int }}_a^{t}exp\left[\omega \left(t-\tau \right)\right]f\left(\tau \right)d\tau ,$$

(65)

where$exp\left[z\right]$is the classical exponential function.

For$\gamma =-1$,$n-\mu =1$, and$\rho =1$, we have the fractional derivative

$$\left({\mathcal{D}}_{1,1,\omega ,a+}^{-1}f\right)\left(t\right)=\underset{a}{\overset{t}{{\displaystyle \int }}}{E}_{1,1}^1\left[\omega \left(t-\tau \right)\right]f^{\left(n\right)}\left(\tau \right)d\tau =\underset{a}{\overset{t}{{\displaystyle \int }}}exp\left[\omega \left(t-\tau \right)\right]f^{\left(n\right)}\left(\tau \right)d\tau ,$$

(66)

where one can consider positive and negative values of$\omega \in ℝ$.

Note that Operators (65) and (66) with$a=0$can be considered in the following form:

$$\omega \left({\epsilon }_{1,1,-\omega ,a+}^{1}f\right)\left(t\right)={{\displaystyle \int }}_0^t\rho \left(\tau \right) f\left(t-\tau \right)d\tau ,$$

(67)

$$\begin{gathered} \omega \left({\mathcal{D}}_{1,1,-\omega ,a+}^{-1}f\right)\left(t\right)={{\displaystyle \int }}_0^t\rho \left(\tau \right) f\left(t-\tau \right)d\tau =\omega {{\displaystyle \int }}_0^{t}exp\left[-\omega \tau \right] f^{\left(n\right)}\left(t-\tau \right)d\tau \\ =\omega {{\displaystyle \int }}_0^{t}exp\left[-\omega \left(t-\tau \right)\right] f^{\left(n\right)}\left(\tau \right)d\tau , \end{gathered}$$

(68)

where$\omega >0,$and$\rho \left(\tau \right)=\omega exp\left[-\omega \tau \right]$is theprobability density function of the exponential distribution.In this case, this operator is interpreted as the derivative of the integer order$n\in \mathbb{N}$, with continuously distributed delay time. The parameter$\left|\omega \right|>0$is often called the rate parameter or the speed of response [47] (p. 27). As an alternative parameter to the speed of response for the exponential lag, one can consider the time constant of this lag that is defined as$T=1/\omega$. This time constant is consistent with the term for the fixed-time delay. For exponentially distributed lag, the parameter$T$is the length of the delay [47] (p. 27). The exponential kernel is actively used in macroeconomic models with distributed lag in the framework of the continuous and discrete-time approaches [47] (p. 26) In economics, the kernel, which is theprobability density function, is called the weighting function [47] (p. 26).

Example 6.

If$\omega =0$, then the kernel of the Prabhakar operator of Caputo type (the D’Ovidio–Polito operator) gives the standard kernel of the Caputo fractional derivative

$$t^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[0\right]=\frac{t^{n-\mu -1}}{\Gamma \left(n-\mu \right)},$$

(69)

and the Caputo fractional derivative

$$\left({\mathcal{D}}_{\rho ,\mu ,0,a+}^{\gamma }f\right)\left(t\right)=\frac{1}{\Gamma \left(n-\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{(t-\tau )}^{n-\mu -1}{f}^{\left(n\right)}\left(\tau \right)d\tau =\left({\mathcal{D}}_{C,a+}^{\mu }f\right)\left(t\right),$$

(70)

where the order is equal to the value$\alpha =\mu >0$.

For$\gamma =0$, we have

$$E_{\rho ,n-\mu }^0\left[\omega t^{\rho }\right]=\frac{1}{\Gamma \left(n-\mu \right)}.$$

(71)

Therefore

$$t^{n-\mu -1}{E}_{\rho ,n-\mu }^0\left[\omega t^{\rho }\right]=\frac{t^{n-\mu -1}}{\Gamma \left(n-\mu \right)},$$

(72)

and also we find the Caputo fractional derivative

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,a+}^{0}f\right)\left(t\right)=\frac{1}{\Gamma \left(n-\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{(t-\tau )}^{n-\mu -1}{f}^{\left(n\right)}\left(\tau \right)d\tau =\left({\mathcal{D}}_{C,a+}^{\mu }f\right)\left(t\right)$$

(73)

with the order$\alpha =\mu >0$.

3. Fractional Differential Equations with Prabhakar Derivatives

3.1. Fractional Differential Equations with the Kilbas–Saigo–Saxena Fractional Derivative

Let us consider the fractional differential equation with the Kilbas–Saigo–Saxena fractional derivative

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)=\frac{d^n}{d{t}^n}\underset{0}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega {\left(t-\tau \right)}^{\rho }\right]f\left(\tau \right)d\tau ,$$

(74)

where $n>\mu ,$ and $\rho ,\mu ,\gamma ,\omega \in ℂ$, $Re\left(\mu \right)>0$, $Re\left(\rho \right)>0$, which contains the Prabhakar function in the kernel. This operator can also be used to describe the economic processes of growth and decay, in which power-law memory and depreciation (or obsolescence) effects are simultaneously manifested.

First, we obtain an expression for the Laplace transform for the KST Operator (74). Using the Laplace transforms (see Equation 1.9.13 in [4] (p. 47), and Equation 5.1.26 in [25] (p. 102)):

$$ℒ\left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left[\omega t^{\rho }\right]\right)\left(s\right)=\frac{s^{-\beta }}{{\left(1-\omega s^{-\alpha }\right)}^{\gamma }},$$

(75)

where $Re\left(s\right)>0$, $Re\left(\beta \right)>0$, $\omega \in ℂ,$ and $\left|\omega s^{-\alpha }\right|<1$, and

$$ℒ\left(f^{\left(n\right)}\left(t\right)\right)\left(s\right)=s^nℒ\left(f\right)\left(s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^k{f}^{\left(n-k-1\right)}\left(0+\right),$$

(76)

we obtain the Laplace transform of the fractional derivative (74) in the following form:

$$\begin{array}{cc}\hfill ℒ\left(\left({\mathit{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)\right)\left(s\right)& =ℒ\left(\frac{d^n}{d{t}^n}\left({\epsilon }_{\rho ,n-\mu ,\omega ,0+}^{-\gamma }f\right)\left(t\right)\right)\left(s\right)\hfill \\ & =s^nℒ\left(\left({\epsilon }_{\rho ,n-\mu ,\omega ,0+}^{-\gamma }f\right)\right)\left(s\right)-{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}{s}^k{\left({\epsilon }_{\rho ,n-\mu ,\omega ,0+}^{-\gamma }f\right)}^{\left(n-k-1\right)}\left(0+\right)\hfill \\ & =s^n\frac{s^{-\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{-\gamma }}ℒ\left(f\right)\left(s\right)-{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}{s}^k\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }f\right)\left(0+\right)\hfill \\ & =\frac{s^{\mu }}{{\left(1-\omega s^{-\rho }\right)}^{-\gamma }}ℒ\left(f\right)\left(s\right)-{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}{s}^k\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }f\right)\left(0+\right),\hfill \end{array}$$

(77)

where we use

$$ℒ\frac{d^n}{d{t}^n}\left({\epsilon }_{\rho ,n-\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\left({\mathit{D}}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)$$

(78)

in the following form:

$$\frac{d^{n-k-1}}{d{t}^{n-k-1}}\left({\epsilon }_{\rho ,n-\mu ,\omega ,a+}^{-\gamma }f\right)\left(t\right)=\frac{d^{n-k-1}}{d{t}^{n-k-1}}\left({\epsilon }_{\rho ,\left(n-k-1\right)-\left(\mu -k-1\right),\omega ,a+}^{-\gamma }f\right)\left(t\right)=\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,a+}^{\gamma }f\right).$$

(79)

Let us present the theorem of the solution of the fractional differential equation with the Kilbas–Saigo–Saxena fractional derivative.

Theorem 3.

Let us consider the fractional differential equation

$$\left({\mathit{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right),$$

(80)

with the Kilbas–Saigo–Saxena fractional derivative, where$y\left(t\right) \in A{C}^n\left[0,b\right]$with$n-1<\mu <n$,$0<t<b\le \infty$, and$\rho ,\mu ,\gamma ,\omega \in ℂ$with$Re\left(\rho \right),Re\left(\mu \right)>0$. Thus, the solution to the Cauchy problem for Equation (80) with initial conditions$y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)}$($k=0,1,\dots n-1$) has the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^{m+1}{t}^{\mu \left(m+1\right)-k-1}{E}_{\rho ,\mu \left(m+1\right)-k}^{\gamma \left(m+1\right)}\left[\omega t^{\rho }\right]\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\rho ,\mu m+\mu }^{\gamma m+\gamma }\left[\omega {\tau }^{\rho }\right]\right\}F\left(t-\tau \right)d\tau , \end{gathered}$$

(81)

where$\mu >n-1,$and$F\left(t\right)$is a function, for which the integral of Equation (81) exists.

Proof. 

Using (77), the Laplace transform of Equation (80) gives

$$\frac{s^{\mu }}{{\left(1-\omega s^{-\rho }\right)}^{-\gamma }}ℒ\left(y\right)\left(s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^k\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right)=\lambda ℒ\left(y\right)\left(s\right)+ℒ\left(F\right)\left(s\right).$$

(82)

Equation (82) can be rewritten as

$$\begin{gathered} ℒ\left(y\right)\left(s\right)-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}{\displaystyle \sum }_{k=0}^{n-1}{s}^k\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right) \\ =\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(y\right)\left(s\right)+\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(F\right)\left(s\right), \end{gathered}$$

(83)

and

$$\begin{gathered} \left(1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)ℒ\left(y\right)\left(s\right) \\ =\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}{\displaystyle \sum }_{k=0}^{n-1}{s}^k\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right)+\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(F\right)\left(s\right). \end{gathered}$$

(84)

This gives rise to the expression of the Laplace transform of the solution

$$\begin{gathered} ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}\frac{\lambda s^{k-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\frac{1}{\left(1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right) \\ +\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\frac{1}{\left(1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}ℒ\left(F\right)\left(s\right). \end{gathered}$$

(85)

This expression can be represented in the following form:

$$\begin{gathered} ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}\frac{\lambda s^{k-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\left({\displaystyle \sum }_{m=0}^{\infty }{\left(\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}^m\right)\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right) \\ +\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\left({\displaystyle \sum }_{m=0}^{\infty }{\left(\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}^m\right)ℒ\left(F\right)\left(s\right), \end{gathered}$$

(86)

where we assume that

$$\left|\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right|<1.$$

(87)

Therefore, we get the following expression:

$$\begin{gathered} ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}\frac{\lambda s^{k-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\left({\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{s}^{-\mu m}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m}}\right)\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right) \\ +\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\left({\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{s}^{-\mu m}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m}}\right)ℒ\left(F\right)\left(s\right), \end{gathered}$$

(88)

and then

$$ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^{m+1}{s}^{-\mu \left(m+1\right)+k}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}}\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right)+{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{s}^{-\mu \left(m+1\right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}}ℒ\left(F\right)\left(s\right).$$

(89)

Then, one can use the expression for the Laplace transform

$$ℒ\left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left[\omega t^{\rho }\right]\right)=\frac{s^{-\beta }}{{\left(1-a{s}^{-\alpha }\right)}^{\gamma }}$$

(90)

in the following form:

$$ℒ\left(t^{\mu \left(m+1\right)-k-1}{E}_{\rho ,\mu \left(m+1\right)-k}^{\gamma \left(m+1\right)}\left[\omega t^{\rho }\right]\right)=\frac{ s^{-\mu \left(m+1\right)+k}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}},$$

(91)

$$ℒ\left(t^{\mu \left(m+1\right)-1}{E}_{\rho ,\mu \left(m+1\right)}^{\gamma \left(m+1\right)}\left[\omega t^{\rho }\right]\right)=\frac{ s^{-\mu \left(m+1\right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}},$$

(92)

where $k=0,1,\dots ,n-1,$ and $\mu -\left(n-1\right)>0$.

As a result, we obtain the following expression:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^{m+1}{t}^{\mu \left(m+1\right)-k-1}{E}_{\rho ,\mu \left(m+1\right)-k}^{\gamma \left(m+1\right)}\left[\omega t^{\rho }\right]\left({\mathit{D}}_{\rho ,\mu -k-1,\omega ,0+}^{\gamma }y\right)\left(0+\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\rho ,\mu m+\mu }^{\gamma m+\gamma }\left[\omega {\tau }^{\rho }\right]\right\}F\left(t-\tau \right)d\tau . \end{gathered}$$

(93)

This ends the proof. □

3.2. Fractional Differential Equation with the D’Ovidio–Polito Operator

Let us consider the fractional differential equation with the D’Ovidio–Polito operator.

Using the Laplace transforms (see Equation 1.9.13 in [4] (p. 47), and Equation 5.1.26 in [25] (p. 102)) in the form

$$ℒ\left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left[\omega t^{\rho }\right]\right)\left(s\right)=\frac{s^{-\beta }}{{\left(1-\omega s^{-\alpha }\right)}^{\gamma }},$$

(94)

where $Re\left(s\right)>0$, $Re\left(\beta \right)>0$, $\omega \in ℂ,$ and $\left|\omega s^{-\alpha }\right|<1$, and

$$ℒ\left(f^{\left(n\right)}\left(t\right)\right)\left(s\right)=s^nℒ\left(f\right)\left(s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^k{f}^{\left(n-k-1\right)}\left(0+\right),$$

(95)

$$ℒ\left(\left({\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)\right)=ℒ\left(t^{n-\mu -1}{E}_{\rho ,n-\mu }^{-\gamma }\left[\omega t^{\rho }\right]\right) ℒ\left(f^{\left(n\right)}\left(t\right)\right),$$

(96)

the Laplace transform of the fractional derivative $\left({\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)$ is given in the following form:

$$ℒ\left(\left({\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }f\right)\left(t\right)\right)\left(s\right)=\frac{s^{-\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{-\gamma }}\left(s^nℒ\left(f\right)\left(s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^k{f}^{\left(n-k-1\right)}\left(0+\right)\right).$$

(97)

Let us consider the fractional differential equation with the D’Ovidio–Polito operator, which contains the Prabhakar function in the kernel and can be used to describe the economic processes of growth and decay, in which power-law memory and depreciation (or obsolescence) effects are simultaneously manifested.

Theorem 4.

Let us consider the fractional differential equation

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right)$$

(98)

with the D’Ovidio–Polito operator, where$y\left(t\right) \in A{C}^n\left[0,b\right]$with$n>\mu$,$0<t<b\le \infty ,$and$\rho ,\mu ,\gamma ,\omega \in ℂ$with$Re\left(\rho \right),Re\left(\mu \right)>0$. Then, the solution of the Cauchy problem for this equation with initial conditions$y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)}$($k=0,1,\dots ,n-1$) has the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{\rho ,\mu m+k+1}^{\gamma m}\left[\omega t^{\rho }\right]y^{\left(k\right)}\left(0+\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\rho ,\mu m+\mu }^{\gamma m+\gamma }\left[\omega {\tau }^{\rho }\right]\right\}F\left(t-\tau \right)d\tau , \end{gathered}$$

(99)

if$F\left(t\right)$is a function for which the integral of Equation (99) exists.

Proof. 

Using (97), the Laplace transform of Equation (98) gives

$$s^{-\left(n-\mu \right)}{\left(1-\omega s^{-\rho }\right)}^{\gamma }\left(s^nℒ\left(y\right)\left(s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^k{y}^{\left(n-k-1\right)}\left(0+\right)\right)=\lambda ℒ\left(y\right)\left(s\right)+ℒ\left(F\right)\left(s\right).$$

(100)

Equation (100) can be rewritten as follows:

$$s^nℒ\left(y\right)\left(s\right)-{\displaystyle \sum }_{k=0}^{n-1}{s}^k{y}^{\left(n-k-1\right)}\left(0+\right)=\frac{\lambda s^{\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(y\right)\left(s\right)+\frac{s^{\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(F\right)\left(s\right).$$

(101)

Therefore, we have

$$\left(s^n-\frac{\lambda s^{\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}{s}^k{y}^{\left(n-k-1\right)}\left(0+\right)+\frac{s^{\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(F\right)\left(s\right).$$

(102)

The Laplace transform $ℒ\left(y\right)\left(s\right)$ of the variable $y\left(t\right)$ is described by the following equation:

$$\begin{gathered} ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}\frac{s^k}{s^n-\frac{\lambda s^{\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}}{y}^{\left(n-k-1\right)}\left(0+\right)+\frac{s^{\left(n-\mu \right)}}{\left(s^n-\frac{\lambda s^{\left(n-\mu \right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right){\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(F\right)\left(s\right)= \\ ={\displaystyle \sum }_{k=0}^{n-1}\frac{s^{k-n}}{1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}}{y}^{\left(n-k-1\right)}\left(0+\right)+\frac{s^{-\mu }}{\left(1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right){\left(1-\omega s^{-\rho }\right)}^{\gamma }}ℒ\left(F\right)\left(s\right) \\ ={\displaystyle \sum }_{k=0}^{n-1}\frac{s^{k-n}}{1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}}{y}^{\left(n-k-1\right)}\left(0+\right)+\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }}ℒ\left(F\right)\left(s\right). \end{gathered}$$

(103)

Let us consider the two terms of this equation separately

$$\frac{s^{k-n}}{1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}}={\displaystyle \sum }_{m=0}^{\infty }{s}^{k-n}{\left(\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}^m={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{s^{-\mu m+k-n}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m}},$$

(104)

$$\begin{gathered} \frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }}=\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\frac{1}{1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}}=\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}{\displaystyle \sum }_{m=0}^{\infty }{\left(\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}^m \\ ={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\left(\frac{s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}^{m+1}={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{ s^{-\mu \left(m+1\right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}}, \end{gathered}$$

(105)

where it is assumed that

$$\left|\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right|<1.$$

(106)

Therefore, Equation (103) takes the following form:

$$\begin{gathered} ℒ\left(y\right)\left(s\right)={\displaystyle \sum }_{k=0}^{n-1}\left({\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{s^{-\mu m+k-n}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m}}\right)y^{\left(n-k-1\right)}\left(0+\right) \\ +\left({\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{ s^{-\mu \left(m+1\right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}}\right)ℒ\left(F\right)\left(s\right) \end{gathered}$$

(107)

Using Equation (94), which describes the Laplace transform of $t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left[\omega t^{\rho }\right]$ in the following form:

$$ℒ\left(t^{\mu m+\left(\mathrm{n}-\mathrm{k}\right)-1}{E}_{\rho ,\mu m+\left(\mathrm{n}-\mathrm{k}\right)}^{\gamma m}\left[\omega t^{\rho }\right]\right)=\frac{ s^{-\mu m+k-n}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m}},$$

(108)

$$ℒ\left(t^{\mu \left(m+1\right)-1}{E}_{\rho ,\mu \left(m+1\right)}^{\gamma \left(m+1\right)}\left[\omega t^{\rho }\right]\right)=\frac{ s^{-\mu \left(m+1\right)}}{{\left(1-\omega s^{-\rho }\right)}^{\gamma \left(m+1\right)}}.$$

(109)

As a result, we obtain the following expression:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\left(n-k\right)-1}{E}_{\rho ,\mu m+\left(n-k\right)}^{\gamma m}\left[\omega t^{\rho }\right]y^{\left(n-k-1\right)}\left(0+\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu \left(m+1\right)-1}{E}_{\rho ,\mu \left(m+1\right)}^{\gamma \left(m+1\right)}\left[\omega {\tau }^{\rho }\right]\right\}F\left(t-\tau \right)d\tau . \end{gathered}$$

(110)

Changing the variable in the first sum ($k\to k^{\prime }=n-k-1$) of Equation (110), we obtain Equation (99).

This ends the proof. □

4. Some Special Cases of the Equation and Its Solutions

Let us consider some corollaries from Theorem 4 for particular cases of the D’Ovidio–Polito operator ${\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{\gamma }$ (the Prabhakar operator of the Caputo type).

Corollary 1.

The fractional differential Equation (98) with$F\left(t\right)=0$and$n=1$($0<\mu <1$) has the following solution:

$$y\left(t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m}{E}_{\rho ,\mu m+1}^{\gamma m}\left[\omega t^{\rho }\right] y_0,$$

(111)

when theinitial condition is$y\left(0\right)=y_0$.

Corollary 2.

The fractional differential Equation (98) with$\gamma =-1$and$\mu =n-1$, has the following form:

$$\left({\mathcal{D}}_{\rho ,n-1,\omega ,0+}^{-1}y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right),$$

(112)

where$n\in \mathbb{N}$, the operator${\mathcal{D}}_{\rho ,1,\omega ,0+}^{-1}$is defined by the following equation:

$$\left({\mathcal{D}}_{\rho ,n-1,\omega ,0+}^{-1}y\right)\left(t\right)=\underset{0}{\overset{t}{{\displaystyle \int }}}{E}_{\rho }\left[\omega {\left(t-\tau \right)}^{\rho }\right]y^{\left(n\right)}\left(\tau \right)d\tau ,$$

(113)

and$E_{\rho }\left[z\right]$is the classical Mittag-Leffler function. Equation (113) has the solution for the case $y\left(0\right)=y_0$in the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\left(n-1\right)m+k}{E}_{\rho ,\left(n-1\right)m+k+1}^{-m}\left[\omega t^{\rho }\right]y^{\left(k\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\left(n-1\right)\left(m+1\right)-1}{E}_{\rho ,\left(n-1\right)\left(m+1\right)}^{-\left(m+1\right)}\left[\omega {\tau }^{\rho }\right]\right\}F\left(t-\tau \right)d\tau , \end{gathered}$$

(114)

where$E_{\rho ,\left(n-1\right)m+k+1}^{-m}\left[\omega t^{\rho }\right]$and$E_{\rho ,\left(n-1\right)\left(m+1\right)}^{-\left(m+1\right)}\left[\omega t^{\rho }\right]$are the$n$-degree polynomials

$$E_{\rho ,\beta }^{-n}\left[\omega t^{\rho }\right]={\displaystyle \sum }_{k=0}^n{\left(-1\right)}^k\left(\begin{array}{c}n\\ k\end{array}\right)\frac{{\omega }^k{t}^{\rho k}}{\Gamma \left(\rho k+\beta \right)}$$

(115)

with$\beta =\left(n-1\right)m+k+1$and$\beta =\left(n-1\right)\left(m+1\right),$respectively.

Let us consider the fractional differential Equation (98) with $\gamma =1$ and $\rho =\mu$.

Corollary 3.

For the case$\gamma =1$and$\rho =\mu$, the fractional differential Equation (98) has the following form:

$$\left({\mathcal{D}}_{\mu ,\mu ,\omega ,0+}^{1}y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right).$$

(116)

The solution of the Cauchy problem for Equation (116) and initial conditions$y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)}$($k=0,1,\dots ,n-1$) is represented in the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}\left\{\frac{t^k}{k!}+\lambda t^{\mu +k}{E}_{\mu ,\mu +k+1}\left[\left(\omega +\lambda \right)t^{\mu }\right]\right\} y^{\left(k\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}{\tau }^{\mu -1}{E}_{\mu ,\mu }\left[\left(\omega +\lambda \right){\tau }^{\mu }\right]F\left(t-\tau \right)d\tau , \end{gathered}$$

(117)

where$n>\mu$and$\mu , \omega \in ℂ$,$Re\left(\mu \right)>0$.

Proof. 

The solution with $\gamma =1$ and $\rho =\mu$ has the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{\mu ,\mu m+k+1}^m\left[\omega t^{\mu }\right]y^{\left(k\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\mu ,\mu m+\mu }^{m+1}\left[\omega {\tau }^{\mu }\right]\right\}F\left(t-\tau \right)d\tau . \end{gathered}$$

(118)

Let us use Equation 1.10.8 from [4] (p. 50) (see also Equation 4.9.6 of [25] (p. 83)), in the following form:

$$t^{\alpha n+\beta -1}{E}_{\alpha ,\alpha n+\beta }^{n+1}\left[\omega t^{\alpha }\right]=\frac{1}{n!}\frac{{\partial }^n}{\partial {\omega }^n}\left(t^{\beta -1}{E}_{\alpha ,\beta }\left[\omega t^{\alpha }\right]\right)$$

(119)

for the first and second terms in (118). Then, we can find the following expressions:

(A)

Using the notations $\alpha =\mu$, $n=m$, $\beta =\mu$ in Equation (118), we find

$$t^{\mu m+\mu -1}{E}_{\mu ,\mu m+\mu }^{m+1}\left[\omega t^{\mu }\right]=\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\mu -1}{E}_{\mu ,\mu }\left[\omega t^{\mu }\right]\right).$$

(120)

In Equation (118), the sum of the second term can be represented in the following form:

$${\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\mu ,\mu m+\mu }^{m+1}\left[\omega {\tau }^{\mu }\right]={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\mu -1}{E}_{\mu ,\mu }\left[\omega t^{\mu }\right]\right).$$

(121)

Using the equation

$$exp\left(\lambda \frac{\partial }{\partial \omega }\right)u\left(\omega \right)=u\left(\omega +\lambda \right),$$

(122)

where $exp\left(\lambda \frac{\partial }{\partial \omega }\right)$ is defined as

$$exp\left(\lambda \frac{\partial }{\partial \omega }\right)u\left(\omega \right)={\displaystyle \sum }_{m=0}^{\infty }\frac{1}{m!}{\left(\lambda \frac{\partial }{\partial \omega }\right)}^{m}u\left(\omega \right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}u\left(\omega \right),$$

(123)

we obtain

$${\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\mu ,\mu m+\mu }^{m+1}\left[\omega {\tau }^{\mu }\right]=\exp \left(\lambda \frac{\partial }{\partial \omega }\right)\left(t^{\mu -1}{E}_{\mu ,\mu }\left[\omega t^{\mu }\right]\right)=t^{\mu -1}{E}_{\mu ,\mu }\left[\left(\omega +\lambda \right)t^{\mu }\right].$$

(124)

As a result, the second term of Equation (118) takes the following form:

$$\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{\mu ,\mu m+\mu }^{m+1}\left[\omega {\tau }^{\mu }\right]\right\}F\left(t-\tau \right)d\tau =\underset{0}{\overset{t}{{\displaystyle \int }}}{\tau }^{\mu -1}{E}_{\mu ,\mu }\left[\left(\omega +\lambda \right){\tau }^{\mu }\right]F\left(t-\tau \right)d\tau .$$

(125)

(B)

Changing the notations $\alpha =\mu$, $n=m$, $\beta =\mu +k+1$ in Equation (120), we get

$$t^{\mu m+\mu +k}{E}_{\mu ,\mu m+\mu +k+1}^{m+1}\left[\omega t^{\mu }\right]=\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\mu +k}{E}_{\mu ,\mu +k+1}\left[\omega t^{\mu }\right]\right).$$

(126)

The sum of the first term can be written as follows:

$$\begin{gathered} {\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{\mu ,\mu m+k+1}^m\left[\omega t^{\mu }\right]=t^k{E}_{\mu ,k+1}^0\left[\omega t^{\mu }\right]+{\displaystyle \sum }_{m=1}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{\mu ,\mu m+k+1}^m\left[\omega t^{\mu }\right] \\ =\frac{t^k}{\Gamma \left(k+1\right)}+{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^{m+1}{t}^{\mu m+\mu +k}{E}_{\mu ,\mu m+\mu +k+1}^{m+1}\left[\omega t^{\mu }\right]. \end{gathered}$$

(127)

Using Equation (126), the sum of terms in (127) can be represented as follows:

$$\begin{gathered} {\displaystyle \sum }_{m=0}^{\infty }{\lambda }^{m+1}{t}^{\mu m+\mu +k}{E}_{\mu ,\mu m+\mu +k+1}^{m+1}\left[\omega t^{\mu }\right]=\lambda {\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\mu +k}{E}_{\mu ,\mu +k+1}\left[\omega t^{\mu }\right]\right) \\ =\lambda exp\left(\lambda \frac{\partial }{\partial \omega }\right) \left(t^{\mu +k}{E}_{\mu ,\mu +k+1}\left[\omega t^{\mu }\right]\right)=\lambda t^{\mu +k}{E}_{\mu ,\mu +k+1}\left[\left(\omega +\lambda \right)t^{\mu }\right]. \end{gathered}$$

(128)

As a result, we obtain

$$\begin{gathered} {\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{\mu ,\mu m+k+1}^m\left[\omega t^{\mu }\right]y^{\left(k\right)}\left(0\right) \\ ={\displaystyle \sum }_{k=0}^{n-1}\left\{\frac{t^k}{k!}+\lambda t^{\mu +k}{E}_{\mu ,\mu +k+1}\left[\left(\omega +\lambda \right)t^{\mu }\right]\right\} y^{\left(k\right)}\left(0\right). \end{gathered}$$

(129)

The use of Equations (125) and (129) in Equation (118) gives Expression (117).

This ends the proof. □

Corollary 4.

In the special case$\gamma =1$and$\rho =\mu <1,$with$n=1$, we have

$$y\left(t\right)=y\left(0\right)+\lambda t^{\mu }{E}_{\mu ,\mu +1}\left[\left(\omega +\lambda \right)t^{\mu }\right]y\left(0\right)+\underset{0}{\overset{t}{{\displaystyle \int }}}{\tau }^{\mu -1}{E}_{\mu ,\mu }\left[\left(\omega +\lambda \right){\tau }^{\mu }\right]F\left(t-\tau \right)d\tau ,$$

(130)

where$n>\mu ,$and$\mu ,\omega \in ℂ$,$Re\left(\mu \right)>0$.

Corollary 5.

For the case$\gamma =1$,$\rho =1,\hspace{1em}\mu =1$with$n=2>\mu$, the fractional differential Equation (98) has the following form:

$$\left({\mathcal{D}}_{1,1,\omega ,0+}^{1}y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right).$$

(131)

The solution of the Cauchy problem for Equation (131) with initial conditions$y\left(0\right)=y_0$and$y^{\left(1\right)}\left(0\right)=y_0^{\left(1\right)}$is represented in the following form:

$$\begin{gathered} y\left(t\right)=y\left(0\right)+\frac{\lambda }{\omega +\lambda }\left(exp\left(\left(\omega +\lambda \right)t\right)-1\right)y\left(0\right)+\lambda t^2{E}_{1,3}\left[\left(\omega +\lambda \right)t\right]y^{\left(1\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}exp\left(\left(\omega +\lambda \right)\tau \right)F\left(t-\tau \right)d\tau . \end{gathered}$$

(132)

Proof. 

For the case $\gamma =1$, $\rho =1,\hspace{1em}\mu =1$, and $n=2>\mu$, Solution (130) has the following form:

$$\begin{gathered} y\left(t\right)=y\left(0\right)+\lambda t E_{1,2}\left[\left(\omega +\lambda \right)t\right]y\left(0\right)+\lambda t^2{E}_{1,3}\left[\left(\omega +\lambda \right)t\right]y^{\left(1\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}{E}_{1,1}\left[\left(\omega +\lambda \right)\tau \right]F\left(t-\tau \right)d\tau . \end{gathered}$$

(133)

Using Equation 1.8.19 from [4] (p. 42), (see also Equation 4.2.1 of [25] (p. 57)) in the following form:

$$E_{1,1}\left[z\right]=\exp \left(z\right),\hspace{1em}{E}_{1,2}\left[z\right]=\frac{\exp \left(z\right)-1}{z},$$

(134)

this solution can be represented by the following equation:

$$\begin{gathered} y\left(t\right)=y\left(0\right)+\frac{\lambda }{\omega +\lambda }\left(\exp \left(\left(\omega +\lambda \right)t\right)-1\right)y\left(0\right)+\lambda t^2{E}_{1,3}\left[\left(\omega +\lambda \right)t\right]y^{\left(1\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\exp \left(\left(\omega +\lambda \right)\tau \right)F\left(t-\tau \right)d\tau , \end{gathered}$$

(135)

where the parameters $\omega$ and $\lambda$ can be positive and negative.

This ends the proof. □

Corollary 6.

The fractional differential Equation (98) with$\omega =0$—that is

$$\left({\mathcal{D}}_{\rho ,\mu ,0,0+}^{\gamma }y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right)$$

(136)

withthe real parameter$\lambda$,$\left(\lambda \in ℝ\right),$and the initial conditions$y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)} \left(k=0,1,\dots ,n-1\right)$has the following solution:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{t}^k{E}_{\mu ,k+1}\left[\lambda t^{\mu }\right]y^{\left(k\right)}\left(0\right)+{{\displaystyle \int }}_0^t{\tau }^{\mu -1}{E}_{\mu ,\mu }\left[\lambda t^{\mu }\right]F\left(t-\tau \right)d\tau .$$

(137)

Proof. 

The fractional differential Equation (98) with $\omega =0$ has the following form:

$$\left({\mathcal{D}}_{\rho ,\mu ,0,a+}^{\gamma }f\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right).$$

(138)

Using the equality

$$\left({\mathcal{D}}_{\rho ,\mu ,0,a+}^{\gamma }f\right)\left(t\right)=\left({\mathcal{D}}_{C,a+}^{\mu }f\right)\left(t\right).$$

(139)

Equation (138) can be represented through the Caputo fractional derivative

$$\left({\mathcal{D}}_{C,0+}^{\mu }y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right).$$

(140)

Then, the solution of the Cauchy problem for Equation (136) takes the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{\rho ,\mu m+k+1}^{\gamma m}\left[0\right]y^{\left(k\right)}\left(0\right) \\ +{{\displaystyle \int }}_0^t\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu \left(m+1\right)-1}{E}_{\rho ,\mu \left(m+1\right)}^{\gamma \left(m+1\right)}\left[0\right]\right\}F\left(t-\tau \right)d\tau . \end{gathered}$$

(141)

Using

$$t^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[0\right]=\frac{t^{\mu -1}}{\Gamma \left(\mu \right)},$$

(142)

we obtain

$$t^{\mu m+k}{E}_{\rho ,\mu m+k+1}^{\gamma m}\left[0\right]=\frac{t^{\mu m+k}}{\Gamma \left(\mu m+k+1\right)},$$

(143)

$$t^{\mu \left(m+1\right)-1}{E}_{\rho ,\mu \left(m+1\right)}^{\gamma \left(m+1\right)}\left[0\right]=\frac{t^{\mu m+\mathsf{\mu }-1}}{\Gamma \left(\mu m+\mathsf{\mu }\right)}.$$

(144)

Substitution of (143) and (144) into (141) gives

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{t^{\mu m+k}}{\Gamma \left(\mu m+k+1\right)}{y}^{\left(k\right)}\left(0\right)+{{\displaystyle \int }}_0^t\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{{\tau }^{\mu m+\mu -1}}{\Gamma \left(\mu m+\mu \right)}\right\}F\left(t-\tau \right)d\tau .$$

(145)

Equation (145) can be rewritten in the following form:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{t}^k\left\{{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{t}^{\mu m}}{\Gamma \left(\mu m+k+1\right)}\right\}{y}^{\left(k\right)}\left(0\right)+\lambda {{\displaystyle \int }}_0^t{\tau }^{\mathsf{\mu }-1}\left\{{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{\tau }^{\mu m}}{\Gamma \left(\mu m+\mathsf{\mu }\right)}\right\}F\left(t-\tau \right)d\tau .$$

(146)

This form allows us to represent the solution through the two-parameter Mittag-Leffler function.

As a result, we obtain the solution in the following form:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{t}^k{E}_{\mu ,k+1}\left[\lambda t^{\mu }\right]y^{\left(k\right)}\left(0\right)+{{\displaystyle \int }}_0^t{\tau }^{\mu -1}{E}_{\mu ,\mu }\left[\lambda t^{\mu }\right]F\left(t-\tau \right)d\tau$$

(147)

which coincides with the standard form of the solution of the fractional differential equation with the Caputo fractional derivative of the order $\alpha =\mu >0$ (see Theorem 5.15 of [4] (p. 323)).

This ends the proof. □

Corollary 7.

The fractional differential Equation (98) with$\gamma =0$in the form

$$\left({\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^{0}y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right),$$

(148)

and the initial conditions$y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)}$($k=0,1,\dots ,n-1$) has the following solution:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{t}^k{E}_{\mu ,k+1}\left[\lambda t^{\mu }\right]y^{\left(k\right)}\left(0\right)+{{\displaystyle \int }}_0^t{\tau }^{\mu -1}{E}_{\mu ,\mu }\left[\lambda t^{\mu }\right]F\left(t-\tau \right)d\tau$$

(149)

since the operator${\mathcal{D}}_{\rho ,\mu ,\omega ,0+}^0$coincides with the Caputo fractional derivative.

Example 7.

In the case $\gamma =0$, $n=1$ with $F\left(t\right)=0$, the solution takes the following form:

$$y\left(t\right)=y\left(0\right){\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{t}^{\mu m}}{\Gamma \left(\mu m+1\right)}=E_{\mu }\left[\lambda t^{\mu }\right]y\left(0\right),$$

(150)

where $E_{\mu }\left[\lambda t^{\mu }\right]$ is the classical Mittag-Leffler function.

5. The Kummer Confluent Hypergeometric Function in the Kernel: Distributed Lag and Memory

Let us consider the special case of $\rho =1$ (see Equation 1.9.3 in [4] (p. 45)), in the following form:

$$E_{1,\mu }^{\gamma }\left[-\omega t\right]=\frac{1}{\Gamma \left(\mu \right)}{F}_{1,1}\left[\gamma ;\mu ;-\omega t\right],$$

(151)

and the fractional integral

$$\left({\epsilon }_{\gamma ,\mu ,-\omega ,a+}f\right)\left(t\right)=\left({\epsilon }_{1,\mu ,-\omega ,a+}^{\gamma }f\right)\left(t\right)=\frac{1}{\Gamma \left(\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{\mu -1}{F}_{1,1}\left[\gamma ;\mu ;-\omega \left(t-\tau \right)\right]f\left(\tau \right)d\tau .$$

(152)

Then, the fractional derivative

$$\left({\mathcal{D}}_{1,\mu ,-\omega ,a+}^{-\gamma }f\right)\left(t\right)=\frac{1}{\Gamma \left(n-\mu \right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{\left(t-\tau \right)}^{n-\mu -1}{F}_{1,1}\left[\gamma ;n-\mu ;-\omega \left(t-\tau \right)\right]f^{\left(n\right)}\left(\tau \right)d\tau ,$$

(153)

where $F_{1,1}\left[a;c;z\right]=\mathsf{\Phi }\left[a;c;z\right]$ is the Kummer hypergeometric function

$$F_{1,1}\left[a;c;z\right]={\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(b\right)}_k}\frac{z^k}{k!}={\displaystyle \sum }_{k=0}^{\infty }\frac{\Gamma \left(a+k\right)\Gamma \left(c\right)}{\Gamma \left(a\right)\Gamma \left(c+k\right)} \frac{z^k}{k!},$$

(154)

where $a, z\in ℂ,$ such that $c\ne 0,-1,-2,\dots$ and Series (154) is absolutely convergent for all $z\in ℂ.$

In general, one can simultaneously take into account lagging and memory phenomena by using the distributed-lag fractional operators proposed in [19] and then applied to macroeconomic models [18]. For example, it is possible to consider the joint effects of the two phenomena: the memory with power-law fading and the lag with gamma distribution of time delay. The memory is described by the Caputo fractional derivatives. The distributed lag is described by the translation operator, in which the delay time $t>0$ is considered to be a random variable that is distributed by probability law (distribution) on a positive semiaxis. The composition of these operators is represented as the Abel-type integral and integro-differential operators with the Kummer confluent hypergeometric function in the kernel.

The Caputo fractional derivative with gamma distributed lag is defined by the following equation:

$$\left({\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\omega ,\mathit{a};\mathbf{\alpha }}f\right)\left(t\right)={{\displaystyle \int }}_0^t{M}_{T }^{\mathbf{\omega },\mathit{a}}\left(\tau \right)\left(D_{C,0+}^{\alpha }f\right)\left(t-\tau \right) d\tau ,$$

(155)

where $M_T\left(\tau \right)=M_{T }^{\omega ,a}\left(\tau \right)$ is the probability density function of the gamma distribution

$$M_{T }^{\mathbf{\omega },\mathit{a}}\left(\tau \right)=\{\begin{array}{c} \frac{{\omega }^a {\tau }^{a-1}}{\Gamma \left(a\right)} exp\left(-\omega \tau \right)\\ 0\end{array}\begin{array}{c}if \tau >0,\\ if \tau \le 0,\end{array}$$

(156)

with the shape parameter $a>0$ and the rate parameter $\omega >0$. If $a=1$, the function (156) describes the exponential distribution. Using the associative property of the Laplace convolution, the Operator (155) can be represented [19] in the following form:

$$\left({\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\mathbf{\omega },\mathit{a};\mathbf{\alpha }}f\right)\left(t\right)={{\displaystyle \int }}_0^t{M}_{TRL }^{\mathbf{\omega },\mathit{a};\mathit{n}-\mathbf{\alpha }}\left(\tau \right)f^{\left(n\right)}\left(t-\tau \right) d\tau ,$$

(157)

where $n-1<\alpha \le n$. Using Equation (151), the kernel $M_{TRL }^{\omega ,a;n-\alpha }\left(t\right)$ can be represented in the following form:

$$M_{TRL }^{\mathbf{\omega },\mathit{a};\mathit{n}-\mathbf{\alpha }}\left(t\right)=\frac{{\omega }^{a }\Gamma \left(a\right)}{\Gamma \left(a+n-\alpha \right)} t^{a+n-\alpha -1}{F}_{1,1}\left[a;a+n-\alpha ;-\omega t\right],$$

(158)

where $F_{1,1}\left(a;b;z\right)$ is the Kummer confluent hypergeometric function (see [42] (p. 115), and [4] (pp. 29–30)).

As a result, we have up to the factor ${\omega }^{a }\Gamma \left(a\right)$ the correspondence between parameters and the operators

$$a=\gamma ,\hspace{1em}\alpha =\mu +\gamma ,$$

(159)

and

$$\left({\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\omega ,\mathit{a};\mathbf{\alpha }}f\right)\left(t\right)={\omega }^{a }\Gamma \left(a\right)\left({\mathcal{D}}_{1,\alpha -a,-\omega ,0+}^{-a}f\right)\left(t\right),$$

(160)

$$\left({\mathcal{D}}_{1,\mu ,-\omega ,0+}^{-\gamma }f\right)\left(t\right)=\frac{1}{{\omega }^{a }\Gamma \left(a\right)}\left({\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\omega ,\mathbf{\gamma };\mathbf{\mu }+\mathbf{\gamma }}f\right)\left(t\right).$$

(161)

This correspondence allows us to provide an interpretation of the parameters of the kernel in the D’Ovidio–Polito operator (the Prabhakar operator of the Caputo type—the regularized Prabhakar fractional derivative). The interpretation of kernel parameters is described by Statement 1.

The Laplace transform of the Caputo fractional derivative with gamma-distributed lag has the following form:

$$\left(ℒ\left({\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\mathbf{\omega },\mathit{a};\mathbf{\alpha }}f\right)\left(t\right)\right)\left(s\right)=\frac{{\omega }^a}{{\left(s+\omega \right)}^a}\left(s^{\alpha }\left(ℒY\right)\left(s\right)-{\displaystyle \sum }_{j=0}^{n-1}{s}^{\alpha -j-1}{f}^{\left(j\right)}\left(0\right)\right),$$

(162)

where $n-1<\alpha \le n$.

As a result, the kernel $M_{TRL }^{\omega ,a;n-\alpha }\left(\tau \right)$ of the proposed special kind of Abel-type fractional derivative describes the joint phenomenon of the power-law fading memory and the continuously distributed lag [19].

As a corollary of Theorem 2, we obtain the following statement. Let us consider the fractional differential equation

$$\left({\mathcal{D}}_{1,\mu ,-\omega ,0+}^{-\gamma }y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right),$$

(163)

with the D’Ovidio–Polito operator, where $y\left(t\right) \in A{C}^n\left[0,b\right]$ with $n>\mu$, $0<t<b\le \infty$, and $\rho ,\mu ,\gamma ,\omega \in ℂ$ with $Re\left(\mu \right)>0$. Then, the solution of the Cauchy problem for this equation with initial conditions $y^{\left(k\right)}\left(0\right)=y_0^{\left(k\right)}$ ($k=0,1,\dots ,n-1$) has the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+k}{E}_{1,\mu m+k+1}^{-\gamma m}\left[-\omega t\right]y^{\left(k\right)}\left(0\right) \\ +\underset{0}{\overset{t}{{\displaystyle \int }}}\left\{{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{\tau }^{\mu m+\mu -1}{E}_{1,\mu m+\mu }^{-\gamma \left(m+1\right)}\left[-\omega \tau \right]\right\}F\left(t-\tau \right)d\tau , \end{gathered}$$

(164)

if $F\left(t\right)$ is a function, for which the integral of Equation (59) exists. This solution can be represented through the Kummer confluent hypergeometric function by using Equation (33) in the following form:

$$\begin{gathered} y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{t}^{\mu m+k}}{\Gamma \left(\mu m+k+1\right)}{F}_{1,1}\left[-\gamma m;\mu m+k+1;-\omega t\right]y^{\left(k\right)}\left(0\right) \\ +{{\displaystyle \int }}_0^t\left\{{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{\tau }^{\mu m+\mu -1}}{\Gamma \left(\mu m+\mu \right)}{F}_{1,1}\left[-\gamma \left(m+1\right);\mu m+\mu ;-\omega \tau \right]\right\}F\left(t-\tau \right)d\tau . \end{gathered}$$

(165)

This solution was obtained in [19] by considering the Caputo fractional derivative with gamma-distributed lag.

Theorem 5.

The fractional differential equation

$$\left({\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\mathbf{\omega },\mathit{a};\mathbf{\alpha }}y\right)\left(t\right)=\lambda y\left(t\right)+F\left(t\right),$$

(166)

where${\mathit{D}}_{\mathit{T};\mathit{C};0+}^{\mathbf{\omega },\mathit{a};\mathbf{\alpha }}$is the fractional derivative of order$\alpha >0$with gamma-distributed lag, in which$a>0$ and$\omega >0$are the shape and rate parameters of the gamma distribution, respectively,has the following solution:

$$y\left(t\right)={\displaystyle \sum }_{j=0}^{n-1}{S}_{\alpha ,a }^{\alpha -j-1}[\lambda {\omega }^{-a},\omega |t]y^{\left(j\right)}\left(0\right)+\frac{1}{\lambda }F\left(t\right)-\frac{1}{\lambda }{{\displaystyle \int }}_0^t{S}_{\alpha ,a }^{\alpha }\left[\lambda {\omega }^{-a},\omega |\tau \right] F\left(t-\tau \right)d\tau ,$$

(167)

where$n=\left[\alpha \right]+1,$and$S_{\alpha ,\delta }^{\gamma }\left[\mu ,\omega |t\right]$is the special function, which is proposed in [19] and defined by the following expression:

$$S_{\alpha ,\delta }^{\gamma }\left[\mu ,\omega |t\right]=-{\displaystyle \sum }_{k=0}^{\infty }\frac{t^{\delta \left(k+1\right)-\alpha k-\gamma -1}}{{\mu }^{k+1}\Gamma \left(\delta \left(k+1\right)-\alpha k-\gamma \right)} F_{1,1}\left[\delta \left(k+1\right);\delta \left(k+1\right)-\alpha k-\gamma ,-\omega t\right],$$

(168)

where$F_{1,1}\left(a;b;z\right)$is the Kummer (confluent hypergeometric) Function (154). Here, we useEquation 5.4.9 of [48,49] in the following form:

$$\left({ℒ}^{-1}\left(\frac{s^a}{{\left(s+b\right)}^c}\right)\right)\left(t\right)=\frac{1}{\Gamma \left(c-a\right)}{t}^{c-a-1}{F}_{1,1}\left(c,c-a,-bt\right),$$

(169)

where$Re\left(c-a\right)>0$.

Proof. 

This theorem is proven in [19]. □

6. Asymptotic Behavior of Depreciation and Obsolescence Processes

6.1. Special Function for Fractional Differential Equations of Depreciation

To simplify discussions of the asymptotic behavior of the obtained solution, we define special functions of one and two variables. Let us give the definition of the function with two variables $x$ and $y$.

Definition 2.

Two-variables V-function is defined by the following equation:

$$V_{\alpha ,\beta ,\delta }^{\mu ,\nu }\left[x,y\right]={\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty }\left(\begin{array}{c}\mu n+\nu +m\\ m\end{array}\right)\frac{x^n{y}^m}{\Gamma \left(\alpha n+\beta m+\delta \right)},$$

(170)

where

$$\left(\begin{array}{c}\mu n+\nu +m\\ m\end{array}\right)=\frac{\Gamma \left(\mu n+\nu +m+1\right)}{\Gamma \left(m+1\right)\Gamma \left(\mu n+\nu +1\right)}=\frac{{\left(\mu n+\nu \right)}_m}{m!}$$

(171)

are the generalized binomial coefficients [4] (pp. 26–27), and${\left(z\right)}_m$is the Pochhammer symbol (see Equations (2) and (3)).

Let us give a representation of this function through the series with the Prabhakar function.

Theorem 6.

The two-variable V-function can be represented through the series with the Prabhakar function by the following equation:

$$V_{\alpha ,\beta ,\delta }^{\mu ,\nu }\left[x,y\right]={\displaystyle \sum }_{n=0}^{\infty }{x}^n E_{\beta ,\alpha n+\delta }^{\mu n+\nu }\left[y\right],$$

(172)

where$E_{\beta ,\alpha n+\delta }^{\mu n+\nu }\left[y\right]$is the Prabhakar function

$$E_{\rho ,\mu }^{\gamma }\left[z\right]={\displaystyle \sum }_{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{\Gamma \left(\rho k+\mu \right)} \frac{z^k}{k!}.$$

(173)

Proof. 

Using the equality

$$\frac{\Gamma \left(\mu n+\nu +m+1\right)}{\Gamma \left(m+1\right)\Gamma \left(\mu n+\nu +1\right)}=\frac{{\left(\mu n+\nu \right)}_m}{m!}$$

(174)

the two-variable V-function can be represented through the series with the Prabhakar function

$$\begin{array}{cc}\hfill V_{\alpha ,\beta ,\delta }^{\mu ,\nu }\left[x,y\right]& ={\displaystyle {\displaystyle \sum }_{n=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{m=0}^{\infty }} \frac{\Gamma \left(\mu n+\nu +m+1\right)}{\Gamma \left(m+1\right)\Gamma \left(\mu n+\nu +1\right)}\frac{x^n{y}^m}{\Gamma \left(\alpha n+\beta m+\delta \right)}\hfill \\ & ={\displaystyle {\displaystyle \sum }_{n=0}^{\infty }}{x}^n{\displaystyle {\displaystyle \sum }_{m=0}^{\infty }} \frac{{\left(\mu n+\nu +1\right)}_m}{m!}\frac{y^m}{\Gamma \left(\alpha n+\beta m+\delta \right)}=\hfill \\ & ={\displaystyle {\displaystyle \sum }_{n=0}^{\infty }}{x}^n{\displaystyle {\displaystyle \sum }_{m=0}^{\infty }} \frac{{\left(\mu n+\nu +1\right)}_m}{\Gamma \left(\beta m+\alpha n+\delta \right)}\frac{y^m}{m!}={\displaystyle {\displaystyle \sum }_{n=0}^{\infty }}{x}^n E_{\beta ,\alpha n+\delta }^{\mu n+\nu +1}\left[y\right].\hfill \end{array}$$

(175)

This ends the proof. □

Using Equation 5.1.25 of [25] or Equation 1.9.12 of [4] (p. 47), the Laplace transform of the Prabhakar function has the following form:

$$ℒ\left(E_{\alpha ,\beta }^{\gamma }\left[\omega t^{\rho }\right]\right)\left(s\right)=\frac{1}{s}{ \mathsf{\Psi }}_{2,1}\left[\begin{array}{c}\left(\gamma ,1\right); \left(1,1\right)\\ \left(\beta ,\alpha \right)\end{array}|\frac{1}{s}\right],$$

(176)

where $Re\left(s\right)>0$, and ${\mathsf{\Psi }}_{2,1}$ is the Wright function. The Laplace transform of the V-function is represented in the following form:

$$ℒ{\left(V_{\alpha ,\beta ,\delta }^{\mu ,\nu }\left[x,y\right]\right)}_y \left(s\right)=\frac{1}{s}{\displaystyle \sum }_{n=0}^{\infty }{ \mathsf{\Psi }}_{2,1}\left[\begin{array}{c}\left(\mu n+\nu ,1\right);\left(1,1\right)\\ \left(\alpha n+\delta ,\beta \right)\end{array}|\frac{1}{s}\right] x^n,$$

(177)

where $Re\left(s\right)>0$.

Let us define the one-variable V-function.

Definition 3.

The one-variable V-function is defined by the following equation:

$$V_{\alpha ,\beta ,\delta }^{\mu ,\nu }\left(a,b,t\right)=t^{\delta -1}{V}_{\alpha ,\beta ,\delta }^{\mu ,\nu -1}\left[a{t}^{\alpha },b{t}^{\beta }\right]={\displaystyle \sum }_{n=0}^{\infty }{\displaystyle \sum }_{m=0}^{\infty } \left(\begin{array}{c}\mu n+\nu -1+m\\ m\end{array}\right)\frac{a^n{b}^m{t}^{\alpha n+\beta m+\delta -1}}{\Gamma \left(\alpha n+\beta m+\delta \right)},$$

(178)

where$Re\left(\beta \right)>0$.

Remark 8.

Note that we use$V_{\alpha ,\beta ,\delta }^{\mu ,\nu -1}\left(a,b,t\right)$instead of$V_{\alpha ,\beta ,\delta }^{\mu ,\nu -1}\left[a{t}^{\alpha },b{t}^{\beta }\right]$in the definition of the one-variable V-function$V_{\alpha ,\beta ,\delta }^{\mu ,\nu }$for simplification. To coordinate the notation in fractional operators, we can change the notation of the indices of the V-function:$m=n$and$k=m$,$\rho =\beta ,$and$\mu =\alpha$,$\gamma =\mu$,$\eta =\mu$.

Then, the one-variable V-function can be represented in the following form:

$$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\rho ,\mu m+\delta }^{\gamma m+\eta }\left[\omega t^{\rho }\right],$$

(179)

where $E_{\alpha ,\beta }^{\gamma }\left[z\right]$ is the Prabhakar function. The one-variable V-function can be written by the following expression:

$$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\displaystyle \sum }_{k=0}^{\infty } {\lambda }^m{\omega }^k\frac{{\left(\gamma m+\eta \right)}_k}{k!}\frac{t^{\rho k+\mu m+\delta -1}}{\Gamma \left(\rho k+\mu m+\delta \right)},$$

(180)

where

$$\frac{{\left(\gamma m+\eta \right)}_k}{k!}=\frac{\Gamma \left(\gamma m+\eta +k\right)}{\Gamma \left(k+1\right)\Gamma \left(\gamma m+\eta \right)}=\left(\begin{array}{c}\gamma m+\eta +k-1\\ k\end{array}\right)$$

(181)

are the generalized binomial coefficients [4].

Theorem 7.

The Laplace transform of the one-variable V-function

$$ℒ\left(V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)\right)=\frac{s^{-\delta }{\left(1-\omega s^{-\rho }\right)}^{-\eta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }},$$

(182)

where the V-function is defined by Equation (178).

Proof. 

To derive the Laplace transform of the one-variable V-function, one can use the following equations:

$$\frac{s^{-\delta }{\left(1-\omega s^{-\rho }\right)}^{-\eta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }}=\frac{s^{-\delta }}{{\left(1-\omega s^{-\rho }\right)}^{\eta }}\frac{1}{1-\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}}=\frac{s^{-\delta }}{{\left(1-\omega s^{-\rho }\right)}^{-\eta }}{\displaystyle \sum }_{m=0}^{\infty }{\left(\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right)}^m,$$

(183)

where it is assumed that

$$\left|\frac{\lambda s^{-\mu }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }}\right|<1.$$

(184)

Therefore, Equation (183) gives

$$\frac{s^{-\delta }{\left(1-\omega s^{-\rho }\right)}^{-\eta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }}={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{ s^{-\mu m-\delta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m+\eta }}.$$

(185)

Let us use the expression for the Laplace transform of Form (94). Using the new variables $\alpha =\rho$, $\beta =\mu m+\delta$, and $\gamma =\gamma m+\eta$, this expression takes the following form:

$$ℒ\left(t^{\mu m+\delta -1}{E}_{\rho ,\mu m+\delta }^{\gamma m+\eta }\left[\omega t^{\rho }\right]\right)=\frac{ s^{-\mu m-\delta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m+\eta }}.$$

(186)

As a result, we obtain the following expression:

$$ℒ\left({\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\rho ,\mu m+\delta }^{\gamma m+\eta }\left[\omega t^{\rho }\right]\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{ s^{-\mu m-\delta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma m+\eta }}.$$

(187)

$$ℒ\left(V_{\rho ,\mathsf{\mu },\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)\right)=\frac{s^{-\delta }{\left(1-\omega s^{-\rho }\right)}^{-\eta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }}.$$

(188)

This ends the proof. □

Remark 9.

Theorem 7 allows us to define the one-variable V-function function by the inverse Laplace transform, as follows:

$$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)={ℒ}^{-1}\left(\frac{s^{-\delta }{\left(1-\omega s^{-\rho }\right)}^{-\eta }}{{\left(1-\omega s^{-\rho }\right)}^{\gamma }-\lambda s^{-\mu }}\right)\left(t\right).$$

(189)

Let us consider the special cases of the one-variable V-function

$$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\rho ,\mu m+\delta }^{\gamma m+\eta }\left[\omega t^{\rho }\right].$$

(190)

Property 1.

For the case$\gamma =1$,$\eta =1,$and$\rho =\mu ,$the one-variable V-function takes the following form:

$$V_{\mu ,\mu ,\delta }^{1,1}\left(\omega ,\lambda ,t\right)=t^{\delta -1}{E}_{\mu ,\delta }\left[\left(\omega +\lambda \right)t^{\mu }\right],$$

(191)

and in the special case$\delta =\mu$, Equation (191) gives the alpha-exponential function

$$V_{\mu ,\mu ,\mu }^{1,1}\left(\omega ,\lambda ,t\right)=e_{\mu }^{\left(\omega +\lambda \right)t}$$

(192)

which is defined (see Equation 1.10.11 in [4] (p. 50)) as follows:

$$e_{\alpha }^{\lambda z}=z^{\alpha -1}{E}_{\alpha ,\alpha }\left[\lambda z^{\alpha }\right].$$

(193)

Proof. 

For $\gamma =1$, $\eta =1,$ and $\rho =\mu$, Equation (190) takes the following form:

$$V_{\mu ,\mu ,\delta }^{1,1}\left(\omega ,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\mu ,\mu m+\delta }^{m+1}\left[\omega t^{\mu }\right].$$

(194)

Using Equation 1.10.8 in [4] (p. 50), (see also Equation 4.9.6 of [25] (p. 83)) in the following form:

$$t^{\alpha m+\beta -1}{E}_{\alpha ,\alpha m+\beta }^{m+1}\left[\omega t^{\alpha }\right]=\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\beta -1}{E}_{\alpha ,\beta }\left[\omega t^{\alpha }\right]\right),$$

(195)

and changing the notations $\alpha =\mu$,$\beta =\delta$, the terms of Sum (194) can be represented as follows:

$$t^{\mu m+\delta -1}{E}_{\mu ,\mu m+\delta }^{m+1}\left[\omega t^{\mu }\right]=\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\delta -1}{E}_{\mu ,\delta }\left[\omega t^{\mu }\right]\right).$$

(196)

Using the equation

$$\exp \left(\lambda \frac{\partial }{\partial \omega }\right)u\left(\omega \right)=u\left(\omega +\lambda \right),$$

(197)

where $\exp \left(\lambda \frac{\partial }{\partial \omega }\right)$ is defined as follows:

$$\exp \left(\lambda \frac{\partial }{\partial \omega }\right)u\left(\omega \right)={\displaystyle \sum }_{m=0}^{\infty }\frac{1}{m!}{\left(\lambda \frac{\partial }{\partial \omega }\right)}^{m}u\left(\omega \right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}u\left(\omega \right),$$

(198)

Equation (194) can be represented in the following form:

$$\begin{gathered} V_{\mu ,\mu ,\delta }^{1,1}\left(\omega ,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\mu ,\mu m+\delta }^{m+1}\left[\omega t^{\mu }\right]={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\delta -1}{E}_{\mu ,\delta }\left[\omega t^{\mu }\right]\right) \\ =\exp \left(\lambda \frac{\partial }{\partial \omega }\right)\left(t^{\delta -1}{E}_{\mu ,\delta }\left[\omega t^{\mu }\right]\right)=t^{\delta -1}{E}_{\mu ,\delta }\left[\left(\omega +\lambda \right)t^{\mu }\right]. \end{gathered}$$

(199)

This ends the proof. □

Property 2.

For the case$\gamma =1$,$\eta =0,$and$\rho =\mu ,$the one-variable V-function takes the following form:

$$V_{\mu ,\mu ,\delta }^{1,0}\left(\omega ,\lambda ,t\right)=\frac{t^{\delta -1}}{\Gamma \left(\delta \right)}+\lambda t^{\mu +\delta -1}{E}_{\mu ,\mu +\delta }\left[\left(\omega +\lambda \right)t^{\mu }\right],$$

(200)

where$\delta >0$, and in the special case

$$V_{\mu ,\mu ,k}^{1,0}\left(\omega ,\lambda ,t\right)=\frac{t^{k-1}}{\left(k-1\right)!}+\lambda t^{\mu +k-1}{E}_{\mu ,\mu +k}\left[\left(\omega +\lambda \right)t^{\mu }\right],$$

(201)

where$k\in \mathbb{N}$.

Proof. 

Let us represent the function in the following form:

$$\begin{gathered} {\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\mu ,\mu m+\delta }^m\left[\omega t^{\mu }\right]=t^{\delta -1}{E}_{\mu ,k+1}^0\left[\omega t^{\mu }\right]+{\displaystyle \sum }_{m=1}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\mu ,\mu m+\delta }^m\left[\omega t^{\mu }\right] \\ =\frac{t^{\delta -1}}{\Gamma \left(\delta \right)}+{\displaystyle \sum }_{m=0}^{\infty }{\lambda }^{m+1}{t}^{\mu m+\mu +\delta -1}{E}_{\mu ,\mu m+\mu +\delta }^{m+1}\left[\omega t^{\mu }\right]. \end{gathered}$$

(202)

Let us use Equation 1.10.8 in [4] (p. 50) (see also equation 4.9.6 of [25] (p. 83)) in the following form:

$$t^{\alpha n+\beta -1}{E}_{\alpha ,\alpha n+\beta }^{n+1}\left[\omega t^{\alpha }\right]=\frac{1}{n!}\frac{{\partial }^n}{\partial {\omega }^n}\left(t^{\beta -1}{E}_{\alpha ,\beta }\left[\omega t^{\alpha }\right]\right).$$

(203)

Changing the notations $\alpha =\mu$, $n=m$, $\beta =\mu +\delta$, we find

$$t^{\mu m+\mu +\delta -1}{E}_{\mu ,\mu m+\mu +\delta }^{m+1}\left[\omega t^{\mu }\right]=\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\mu +\delta -1}{E}_{\mu ,\mu +\delta }\left[\omega t^{\mu }\right]\right).$$

(204)

Using Equations (197) and (198), the sum in Equation (202) can be written in the following form:

$$\begin{gathered} {\displaystyle \sum }_{m=0}^{\infty }{\lambda }^{m+1}{t}^{\mu m+\mu +\delta -1}{E}_{\mu ,\mu m+\mu +\delta }^{m+1}\left[\omega t^{\mu }\right]=\lambda {\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{1}{m!}\frac{{\partial }^m}{\partial {\omega }^m}\left(t^{\mu +\delta -1}{E}_{\mu ,\mu +\delta }\left[\omega t^{\mu }\right]\right) \\ =\lambda \exp \left(\lambda \frac{\partial }{\partial \omega }\right) \left(t^{\mu +\delta -1}{E}_{\mu ,\mu +\delta }\left[\omega t^{\mu }\right]\right)=\lambda t^{\mu +\delta -1}{E}_{\mu ,\mu +\delta }\left[\left(\omega +\lambda \right)t^{\mu }\right]. \end{gathered}$$

(205)

As a result, Equation (205) gives

$${\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\mu ,\mu m+\delta -1}^m\left[\omega t^{\mu }\right]=\frac{t^{\delta -1}}{\Gamma \left(\delta \right)}+\lambda t^{\mu +\delta -1}{E}_{\mu ,\mu +\delta }\left[\left(\omega +\lambda \right)t^{\mu }\right].$$

(206)

This ends the proof. □

Property 3.

For$\omega =0$,the function$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)$is represented through the classical Mittag-Leffler function by the following expression:

$$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(0,\lambda ,t\right)=t^{\delta -1}{E}_{\mu ,\delta }\left[\lambda t^{\mu }\right].$$

(207)

Proof. 

Using

$$t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left[0\right]=\frac{t^{\beta -1}}{\Gamma \left(\beta \right)},$$

(208)

Equation (190) gives

$$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(0,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\rho ,\mu m+\delta }^{\gamma m+\eta }\left[0\right]={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{t^{\mu m+\delta -1}}{\Gamma \left(\mu m+\delta \right)}=t^{\delta -1}{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{t}^{\mu m}}{\Gamma \left(\mu m+\delta \right)}=t^{\delta -1}{E}_{\mu ,\delta }\left[\lambda t^{\mu }\right].$$

(209)

This ends the proof. □

Property 4.

For$\gamma =\eta =0$,the function$V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)$is represented through the classical Mittag-Leffler function

$$V_{\rho ,\mu ,\delta }^{0,0}\left(\omega ,\lambda ,t\right)=t^{\delta -1}{E}_{\mu ,\delta }\left[\lambda t^{\mu }\right].$$

(210)

Proof. 

Using

$$E_{\alpha ,\beta }^0\left[z\right]=\frac{1}{\Gamma \left(\beta \right)},$$

(211)

we find that

$$\begin{gathered} V_{\rho ,\mu ,\delta }^{0,0}\left(0,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\rho ,\mu m+\delta }^0\left[\omega t^{\rho }\right]={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m\frac{t^{\mu m+\delta -1}}{\Gamma \left(\mu m+\delta \right)} \\ =t^{\delta -1}{\displaystyle \sum }_{m=0}^{\infty }\frac{{\lambda }^m{t}^{\mu m}}{\Gamma \left(\mu m+\delta \right)}=t^{\delta -1}{E}_{\mu ,\delta }\left[\lambda t^{\mu }\right]. \end{gathered}$$

(212)

This ends the proof. □

Using the V-function, the solution of the Cauchy problem for Equation (98) is represented through this special function in the following form:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{V}_{\rho ,\mu ,k+1}^{\gamma ,0}\left(\omega ,\lambda ,t\right)y^{\left(k\right)}\left(0\right)+\underset{0}{\overset{t}{{\displaystyle \int }}}{V}_{\rho ,\mu ,\mu }^{\gamma ,\gamma }\left(\omega ,\lambda ,\tau \right)F\left(t-\tau \right)d\tau .$$

(213)

As a result, the asymptotic behavior of the solution is determined by the asymptotics of the function $V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)$. The asymptotic behavior ($t\to \infty$) of the solution with $F\left(t\right)=0$ defines the warranted growth rate in economics. This behavior is determined by the asymptotics of the one-variable V-function in the following form:

$$V_{\rho ,\mu ,k+1}^{\gamma ,0}\left(\omega ,\lambda ,t\right)={\displaystyle \sum }_{m=0}^{\infty }{\lambda }^m{t}^{\mu m+\delta -1}{E}_{\rho ,\mu m+k+1}^{\gamma m}\left[\omega t^{\rho }\right].$$

(214)

The warranted growth rate of economic models that is described by Equation (59) is defined by the asymptotic properties of the function $V_{\rho ,\mu ,k+1}^{\gamma ,0}\left(\omega ,\lambda ,t\right)$.

An accurate description of the asymptotic behavior $V_{\rho ,\mu ,\delta }^{\gamma ,\eta }\left(\omega ,\lambda ,t\right)$ is not currently described in the general case.

6.2. Some Properties of Asymptotic Behavior and Special Case

Obviously, the asymptotic behavior of the function $V_{\rho ,\mu ,\beta }^{\gamma }\left(\omega ,\lambda ,t\right)$ depends on the asymptotic of the Prabhakar functions $E_{\rho ,\mu m+\left(n-k\right)}^{\gamma m}\left[\omega t^{\rho }\right]$. This asymptotic is described by Paris in the handbook [46] (pp. 297–325). Using Equations 44–47 in [46] (pp. 312–313), one can see that the asymptotic expressions contain the factors in the form of the following exponents:

$$\exp \left({\left(\omega t^{\rho }\right)}^{1/\rho }\right)=\exp \left({\omega }^{1/\rho }t\right)$$

(215)

with $t>0$, which define the warranted growth rate in economics. This factor is common to all functions $E_{\rho ,\mu m+\left(\mathrm{n}-\mathrm{k}\right)}^{\gamma m}\left[\omega t^{\rho }\right]$, and does not depend on the values of $m=0,1,2,\dots$ As a result, the warranted growth rate depends on the memory parameters $\omega$ and $\rho$. Therefore, one can state that the effective growth rate for processes with memory described by the Prabhakar function can be considered as a function in the following form:

$${\lambda }_{eff}\left(\rho ,\mu ,\gamma ,\omega ,\lambda \right)=f\left(\lambda ,\mu ,\gamma ,{\omega }^{1/\rho }\right).$$

(216)

The dependence on the parameter $\lambda$ can be established only upon receipt of the expression of the asymptotics of the function $V_{\rho ,\mu ,\mathrm{k}}^{\gamma ,n}\left(\omega ,\lambda ,t\right)$.

It should be emphasized that the Caputo fractional derivative is a special case of the Prabhakar fractional derivative for $\omega =0$. In [18], we proved that with the use of the Caputo fractional derivative in the linear fractional differential growth equation ($\lambda >0$), the warranted growth rate with memory is described by the following equations:

$${\lambda }_{eff}\left(\rho ,\mu ,\gamma ,0,\lambda \right)={\lambda }^{1/\mu },$$

(217)

$${\lambda }_{eff}\left(\rho ,\mu ,0,\omega ,\lambda \right)={\lambda }^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.}.$$

(218)

Let us prove this statement. The use of Equation (98) with $F\left(t\right)=0$ and $\mu =\alpha$ allows us to obtain the warranted growth rate for processes with power-law fading memory [18]. Equation (98) has the solution (see Theorem 5.15 of [4] (p. 323)) in the following form:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{n-1}{y}^{\left(k\right)}\left(0\right)t^k{E}_{\alpha ,k+1}\left[\lambda t^{\alpha }\right],$$

(219)

where $y^{\left(k\right)}\left(0\right)$ represents the integer-order derivatives of orders $k\ge 0$ at $t=0,$ where $n-1<\alpha \le n$ and $n\in \mathbb{N}$. Solution (219) is expressed through the two-parameter Mittag-Leffler function [24].

In the case $0<\alpha \le 1$($n=1$), Equation (219) takes the following form:

$$y\left(t\right)=y\left(0\right)E_{\alpha ,1}\left[\lambda t^{\alpha }\right].$$

(220)

For $1<\alpha \le 2$($n=2$), Equation (219) gives

$$y\left(t\right)=y\left(0\right) E_{\alpha ,1}\left[\lambda t^{\alpha }\right]+y^{\left(1\right)}\left(0\right) t E_{\alpha ,2}\left[\lambda t^{\alpha }\right],$$

(221)

where $y^{\left(1\right)}\left(0\right)$ is first-order derivative of $y\left(t\right)$ at $t=0$.

If $\alpha =1$, then we get the following solution:

$$y\left(t\right)=y\left(0\right) \exp \left(\lambda t\right),$$

(222)

where the value $\lambda$ describes the rate of decline or growth without memory. In economics, the value $\lambda >0$ is called the warranted growth rate [47] (p. 67).

Let us consider the asymptotic behavior of the solution at $t\to \infty$ by using the asymptotic equation of the function $E_{\alpha ,k+1}\left[\lambda t^{\mathsf{\alpha }}\right]$ at $t\to \infty$.

For $0<\alpha <2,$ Equation 1.8.27 of [4] (p. 43) can be used in the following form:

$$E_{\alpha ,\beta +1}\left[z\right]=\frac{z^{-\beta /\alpha }}{\alpha }\exp \left(z^{1/\alpha }\right)-{\displaystyle \sum }_{k=1}^m\frac{1}{\Gamma \left(\beta +1-\alpha k\right)}\frac{1}{z^k}+O\left(\frac{1}{z^{m+1}}\right),$$

(223)

where $\beta \in ℂ,$ at $\left|z\right|\to \infty$ and $\left|\arg \left(z\right)\right|\le \mu$. Here, $\mu$ is a real number ($\mu \in ℝ$) such that $\pi \alpha /2<\mu <\inf \left\{\pi ,\pi \alpha \right\},$ and all $m\in \mathbb{N}$. For $\mu \le \left|\arg \left(z\right)\right|\le \pi$, we have

$$E_{\alpha ,\beta +1}\left[z\right]=-{\displaystyle \sum }_{k=1}^m\frac{1}{\Gamma \left(\beta +1-\alpha k\right)}\frac{1}{z^k}+O\left(\frac{1}{z^{m+1}}\right)$$

(224)

for $\left|z\right|\to \infty$ and all $m\in \mathbb{N}$. Equations (223) and (224) describe the asymptotic behavior at infinity for the case $0<\alpha <2$.

For $0<\alpha <2,$ using Equation (223) with $z=\lambda t^{\alpha }$, we get the following equation:

$$E_{\alpha ,\beta +1}\left[\lambda t^{\alpha }\right]=\frac{{\lambda }^{-\beta /\alpha } t^{-\beta }}{\alpha }\exp \left({\lambda }^{1/\alpha }t\right)-{\displaystyle \sum }_{j=1}^m\frac{{\lambda }^{-j}}{\Gamma \left(\beta +1-\alpha j\right)}\frac{1}{t^{\alpha ·j}}+O\left(\frac{1}{t^{\alpha \left(m+1\right)}}\right)$$

(225)

for $t\to \infty$, where $\lambda >0$ is a real number. The asymptotic Equation (225) allows us to describe the behavior at $t\to \infty$ in a processes power-law memory, in which the memory fading parameter is $0<\mathsf{\alpha }<2$. Substitution of Expression (225) with $\lambda >0$ and $\beta =k$ into (219) gives

$$\begin{array}{ll}y\left(t\right)=\exp \left({\lambda }^{1/\alpha } t\right)& {\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}{y}^{\left(k\right)}\left(0\right)\frac{{\lambda }^{-k/\alpha }}{\alpha }\\ & +{\displaystyle {\displaystyle \sum }_{k=0}^{n-1}}\left({\displaystyle {\displaystyle \sum }_{j=1}^m}\frac{y^{\left(k\right)}\left(0\right){\lambda }^{-j}}{\Gamma \left(k+1-\alpha j\right)} t^{k-\alpha j}+O\left(\frac{1}{t^{\alpha ·\left(m+1\right)-k}}\right)\right),\end{array}$$

(226)

where $0<n-1<\alpha <n$. Expression (226) describes the behavior of Solution (219) at $t\to \infty$. For the non-integer values of the fading parameter $0<\alpha <2$, the behavior of process $y\left(t\right)$ with memory is determined by the term with $\exp \left({\lambda }^{1/\alpha } t\right)$. The power-law terms $t^{k-\alpha j}$ of (226) do not determine the dominant behavior at $t\to \infty$ if

$${\displaystyle \sum }_{k=0}^{n-1}{y}^{\left(k\right)}\left(0\right)\frac{{\lambda }^{-k/\alpha }}{\alpha }\ne 0.$$

(227)

As a result, one can state that the effective growth rate for processes with memory, which is described by the Prabhakar function, has the following property:

$${\lambda }_{eff}\left(\rho ,\mu ,\gamma ,0,\lambda \right)={\lambda }^{1/\mu }.$$

(228)

This statement can be proven similarly. These statements can be interpreted as the correspondence principle.

As a result, we formulated the principle of changing warranted (technological) growth rate by memory [18].

Principle of changing of warranted growth rates:

The memory fading parameter($\mu =\rho$) can both increase and decrease the warranted growth rate ($\lambda$) of economy according to the following equation:

$${\lambda }_{eff}={\lambda }^{1/\mu }.$$

(229)

For small technological growth rates, which are described by the standard model, the effects of one-parameter memory with the fading parameter$0<\alpha <1$ lead to decrease in the growth rates of the economy, and lead to an increase in the growth rates for$\alpha >1$. For the large rates of technological growth that are described by the standard model, the effects of power-law memory with the fading$0<\alpha <1$ lead to an increase in the growth rates of the economy, and lead to a decrease in growth rates for$\alpha >1$.

Let us consider the asymptotic behavior for the special case $\gamma =1$ and $\rho =\mu$. For $0<\alpha <2,$ using Equation (18) with $z=\lambda t^{\alpha }$, we get the following equation:

$$t^{\beta }{E}_{\alpha ,\beta +1}\left[\lambda t^{\alpha }\right]=\frac{{\lambda }^{-\beta /\alpha }}{\alpha }\exp \left({\lambda }^{1/\alpha }t\right)-{\displaystyle \sum }_{j=1}^m\frac{{\lambda }^{-j}}{\Gamma \left(\beta +1-\alpha j\right)}{t}^{\beta -\alpha j}+O\left(\frac{1}{t^{\alpha \left(m+1\right)}}\right)$$

(230)

for $t\to \infty$, where $\lambda >0$ is a real number. Then, we get the asymptotic expression for the case $n=1$ and $F\left(t\right)=0$, in the following form:

$$y\left(t\right)=\lambda t^{\mu }{E}_{\mu ,\mu +1}\left[\left(\omega +\lambda \right)t^{\mu }\right]y\left(0\right)=\frac{1}{\alpha }\exp \left({\left(\omega +\lambda \right)}^{1/\mu }t\right)-{\displaystyle \sum }_{j=1}^m\frac{{\lambda }^{1-j}}{\Gamma \left(\mu +1-\mu j\right)}{t}^{\mu -\mu j}+O\left(t^{\mu -\mu \left(m+1\right)}\right).$$

(231)

These results allow us to state that

$${\lambda }_{eff}\left(\mu ,\mu ,1,\omega ,\lambda \right)={\left(\omega +\lambda \right)}^{1/\mu },$$

(232)

and in the special case

$${\lambda }_{eff}\left(1,1,1,\omega ,\lambda \right)=\omega +\lambda .$$

(233)

Note that this solution can be considered for negative values of $\lambda$ or $\omega$, such that $\left(\omega +\lambda \right)>0$. This can also include the special cases when $\omega <0$, such that $\omega +\lambda =\lambda -\left|\omega \right|>0$, and the case λ $<0$, such that $\omega +\lambda =\omega -\left|\lambda \right|>0$.

As a result, one can formulate the following principle of changing of warranted growth rates for the case of memory that is described by the generalization of the exponential function (the two-parameter Mittag-Leffler function).

Principle of changing of warranted growth rates:

The rate of change of memory($\omega$) and the memory fading parameter ($\mu =\rho$) can both increase and decrease the warranted growth rate ($\lambda$) of the economy according to the following equation:

$${\lambda }_{eff}={\left(\omega +\lambda \right)}^{1/\mu },$$

(234)

where the parameters$\lambda$or$\omega$can be positive andnegative, such that$\omega +\lambda >0$in general.

7. Conclusions

In this paper, the integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function as the kernels are considered. An important distinguishing feature of these operator kernels is the fact that such operators can be used to describe non-exponential depreciation and memory in economics. Equations with the Prabhakar operator of the Riemann–Liouville and Caputo types are considered. The solutions of these fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed.

Note that the most common fractional integrals and derivatives (primarily Riemann–Liouville, Hadamard, Riesz, Caputo, Grunwald–Letnikov, Erdelyi–Kober) cannot be used to describe the depreciation or aging phenomena in economics. This fact is due to the properties of the operator kernels of these fractional operators of non-integer order. The proposed article proves distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics. The aging and depreciation phenomena can be described by using the fractional derivatives and integrals with the Prabhakar function, the Kummer confluent hypergeometric function, and the hypergeometric function in the kernels. These functions were first used as kernels of fractional integrals more than 40 years ago in [23,50,51] by Prabhakar, (see also [1] (pp. 731–737)). The fractional derivatives of the Riemann–Liouville type with the Prabhakar function in the kernels were proposed by Kilbas, Saigo, and Saxena in 2004 [27], and the fractional derivatives of the Caputo type with the Prabhakar function in the kernels were proposed by D’Ovidio and Polito in 2013 [37,38,39]. The Kummer confluent hypergeometric function in the kernels of integral and integro-differential operators allows us to describe the joint effects of two phenomena: the fading memory and the distributed lag [18,19,45].

This article describes the main distinguishing features of the fractional derivatives and integrals with the Prabhakar function in the kernels. It was proven that these operators allow us to take into account the joint effects of two phenomena: non-exponential depreciation and fading memory. Exact solutions of the fractional differential equations with the fractional derivatives of the Prabhakar type are derived in this paper. These solutions can be used to describe fractional dynamic models with memory, depreciation, and obsolescence in economics, physics, biology, and social and other sciences.