**1. Introduction**

In fractional calculus, the standard calculus operations of differentiation and integration are generalised to orders beyond the integers: rational, real, and even complex numbers can be used for the order of *differintegration* [1–3]. This area of research is four centuries old, but it has expanded rapidly only in the last fifty years, discovering applications in many fields of science and engineering [4–6]. The most commonly used definition of fractional derivatives and integrals is the Riemann–Liouville one, where fractional integrals are defined by

$$\prescript{RL}{a}{D}\_{x}^{a}f(x) = \frac{1}{\Gamma(-a)} \int\_{a}^{x} (x - y)^{-a-1} f(y) \, \mathrm{d}y, \qquad \mathrm{Re}(a) < 0,\tag{1}$$

and fractional derivatives are defined by

$${}^{RL}\_{a}D\_{x}^{a}f(\mathbf{x}) = \frac{\mathbf{d}^{m}}{\mathbf{d}\mathbf{x}^{m}} \Big( {}^{RL}\_{a}D\_{x}^{a-m}f(\mathbf{x}) \Big), \qquad \text{Re}(a) \ge 0, m := \lfloor \text{Re}(a) \rfloor + 1. \tag{2}$$

Here, *D<sup>α</sup> f* denotes the derivative to order *α* of a function *f* , and *a* is a constant of differintegration. It is important to note that in fractional calculus, derivatives as well as integrals rely on the choice of an arbitrary constant *a*. This constant is usually set to be either *a* = 0 or *a* = −∞. To see why both

choices are useful, we present the following Lemma which provides two "'natural" differintegration formulae, one requiring *a* = 0 and the other requiring *a* = −∞. Neither option can be eliminated from the range of possible values for *a*, if we wish to retain natural expressions for differintegrals of elementary functions.

**Lemma 1.** *The Riemann–Liouville (RL) differintegrals of power functions and exponential functions, with constant of differintegration a* = 0 *and a* = −∞*, respectively, are as follows.*

$$\mathbf{x}^{\text{RL}}{}\_{0}D\_{\mathbf{x}}^{\text{a}}(\mathbf{x}^{\emptyset}) = \frac{\Gamma(\beta + 1)}{\Gamma(\beta - \alpha + 1)} \mathbf{x}^{\beta - \alpha}, \qquad \mathbf{a}, \beta \in \mathbb{C}, \text{Re}(\beta) > -1; \tag{3}$$

$$D\_{-\\\infty}^{RL}D\_x^{\mathfrak{a}}\left(e^{\mathfrak{f}\cdot\mathbf{x}}\right) = \beta^{\mathfrak{a}}e^{\mathfrak{f}\cdot\mathbf{x}}, \qquad \mathfrak{a}, \beta \in \mathbb{C}, \beta \notin \mathbb{R}\_0^-. \tag{4}$$

*In both cases, we define complex power functions using the principal branch with arguments between* −*π and π.*

**Proof.** In both cases, the proof for fractional integrals follows from manipulation and substitution in the integral formula in Equation (1), and then the proof for fractional derivatives is immediate from the definition in Equation (2). For more details, we refer the reader to [1,3].

In recent years, many alternative definitions of fractional differintegrals have been proposed. Some of these were motivated by the different real-world systems which can be modelled by different fractional-calculus structures: for example, replacing the power function in Equation (1) by another function to better describe certain types of processes in dynamical systems [7,8]. Others were created by adding extra parameters and levels of generalisation into functions and formulae [9–11].

One particular function which frequently appears in the study of fractional derivatives and integrals [12–14] is the **Mittag–Leffler function**, which in its simplest form is defined by

$$E\_n(z) = \sum\_{n=0}^{\infty} \frac{z^n}{\Gamma(n\alpha + 1)}, \qquad z \in \mathbb{C}.\tag{5}$$

The above function with a single parameter *α* has also been extended to more general functions defined with two or more parameters, such as the following [15,16].

$$E\_{a,\beta}(z) = \sum\_{n=0}^{\infty} \frac{z^n}{\Gamma(na + \beta)}, \qquad z \in \mathbb{C};\tag{6}$$

$$E\_{a,\beta}^{\rho}(z) = \sum\_{n=0}^{\infty} \frac{\Gamma(\rho + n)z^n}{\Gamma(\rho)\Gamma(n\alpha + \beta)n!}, \qquad z \in \mathbb{C};\tag{7}$$

$$E\_{a,\beta}^{\rho,\mathbf{x}}(z) = \sum\_{n=0}^{\infty} \frac{\Gamma(\rho + \kappa n) z^n}{\Gamma(\rho) \Gamma(n\kappa + \beta) n!}, \qquad z \in \mathbb{C}.\tag{8}$$

It is clear that the following interrelations hold between the above functions:

$$E\_{a,\emptyset}^{\rho,1}(z) = E\_{a,\emptyset}^{\rho}(z), \qquad E\_{a,\emptyset}^1(z) = E\_{a,\emptyset}(z), \qquad E\_{a,1}(z) = E\_a(z).$$

Several new models of fractional calculus have used such functions in their definitions, and we mention two of these in particular.

The **AB model**, formulated by Atangana and Baleanu [7] and further studied in [17–20], is defined by replacing the power function in Equation (1) by a one-parameter Mittag–Leffler function of the type in Equation (5):

$${}^{ABR}{}\_{d}D\_{x}^{a}f(x) = \frac{B(a)}{1-a} \cdot \frac{d}{dx} \int\_{a}^{x} \mathbb{E}\_{a} \left( \frac{-a}{1-a} (x-y)^{a} \right) f(y) \, \mathrm{d}y, \qquad 0 < a < 1,\tag{9}$$

$$I\_{a}^{AB}I\_{x}^{a}f(\mathbf{x}) = \frac{1-a}{B(a)}f(\mathbf{x}) + \frac{a}{B(a)}\, \, ^{RL}\_{a}I\_{x}^{a}f(\mathbf{x}), \tag{10}$$

We note that here *α* is a real variable and not a complex one. All discussion of the AB model in the literature so far has assumed the order of differentiation to be real. The first paper to consider complex-order AB differintegrals is currently in press [21].

The **Prabhakar model**, based on an integral operator defined in 1971 [22] but only later formulated as part of fractional calculus [23,24], is defined by replacing the power function in Equation (1) by a three-parameter Mittag–Leffler function of the type in Equation (7). This model has also been generalised [16] to use a 4-parameter Mittag–Leffler function of the type in Equation (8), and its properties have been explored in many papers (e.g., [25–27]).

One useful application of fractional calculus in pure mathematics has been to find new functional equations and interrelations between various important functions. For example, fractional versions of the product rule and chain rule have given rise to new formulae for assorted special functions [28,29], and fractional differintegration of infinite series has yielded new identities on the Riemann zeta function and its generalisations [30–32].

In the current work, we use these techniques to prove new relationships between the several Mittag–Leffler functions defined above. It is possible to write the three-parameter Mittag–Leffler function as a fractional differintegral of the two-parameter one, as well as writing the two-parameter one as a fractional differintegral of the one-parameter one, and thence to deduce an integral relationship between the one-parameter and three-parameter Mittag–Leffler functions, which suggests a relationship between the AB and Prabhakar models of fractional calculus. Throughout all of this, we use only the classical Riemann–Liouville fractional integrals and derivatives. We also examine the possibility of applications of these results in fields of science such as bioengineering and dielectric relaxation.

This paper is structured as follows. In Section 2, through a number of theorems and propositions, we state the main results concerning relationships between Mittag–Leffler functions. In Section 3, we discuss applications, and, in Section 4, we conclude the article.

#### **2. The Main Results**

We first state an important result about fractional differintegration of series, which we need to use in the proofs below.

**Lemma 2.** *Consider a function S defined by an infinite series*

$$S(\mathfrak{x}) = \sum\_{n=1}^{\infty} S\_n(\mathfrak{x})$$

*which is uniformly convergent on the set* <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>a</sup>*| ≤ *<sup>K</sup> for some fixed constants <sup>a</sup>* <sup>∈</sup> <sup>C</sup>*, <sup>K</sup>* <sup>&</sup>gt; <sup>0</sup>*. Let <sup>α</sup>* <sup>∈</sup> <sup>C</sup> *be a fixed order of differintegration.*

*1. If* Re(*α*) < 0 *(fractional integration), then we have*

$$\prescript{RL}{a}{D}\_{x}^{a}\mathcal{S}(x) = \sum\_{n=1}^{\infty} \prescript{RL}{a}{D}\_{x}^{a}\mathcal{S}\_{n}(x), \qquad |x - a| \le K\_{\prime}$$

*and the series on the right-hand side is uniformly convergent on the given region.*

*2. If* Re(*α*) <sup>≥</sup> <sup>0</sup> *(fractional differentiation) and the series* <sup>∑</sup><sup>∞</sup> *n*=1 *RL aD<sup>α</sup> xSn*(*x*) *is uniformly convergent on the region* |*x* − *a*| ≤ *K, then we have*

$$\prescript{RL}{a}{D}\_{x}^{\mathfrak{a}}\mathcal{S}(x) = \sum\_{n=1}^{\infty} \prescript{RL}{a}{D}\_{x}^{\mathfrak{a}}\mathcal{S}\_{n}(x), \qquad |x - a| \le K.$$
