A Distributional Framework Based on Gamma–Zeta Operators for Singular Fractional Models

Abstract

Fractional calculus and distribution theory share a common conceptual origin in the symbolic interpretation of differentiation and integration. Despite this connection, most developments in fractional calculus have traditionally been formulated within the framework of ordinary functions, while the systematic use of distributions remains limited. In this work, a novel distributional framework is developed by constructing a fractional Taylor representation of the product of Euler gamma and Riemann zeta functions in terms of fractional derivatives of the Dirac delta distribution. The proposed formulation enables the derivation of new fractional identities via Laplace transformation and facilitates the analytical solution of fractional differential equations containing such functions. Closed-form solutions are obtained in both classical and generalized distributional senses, allowing the extension of solutions from the positive real axis to the entire real line. Furthermore, the framework is applied to fractional operators of Erdélyi–Kober type, yielding new integral and derivative transforms. Fractional differential and integral equations with singular terms arise naturally in several engineering models involving memory effects, impulsive responses, and anomalous transport phenomena. However, the presence of nonremovable singularities—such as those associated with Euler gamma and Riemann zeta functions—significantly restricts the applicability of classical analytical methods. Overall, the proposed distributional framework bridges the gap between abstract fractional calculus and practical engineering models by enabling analytical solutions of fractional systems with singular memory kernels that were previously inaccessible using classical methods.

1. Introduction and Motivation

Leibniz’s original interpretation of integration as symbolic negative powers of differentiation played a fundamental role in the early development of fractional calculus [1,2]. Similarly, the theory of distributions (generalized functions) emerged from the need to provide a rigorous framework for symbolic manipulations frequently used in engineering and physics [3]. This shared conceptual origin suggests a natural relationship between fractional calculus and distribution theory. Despite this connection, most studies in fractional calculus continue to focus primarily on ordinary functions (see ref. [4] and the references therein).

The integration of distributions into fractional calculus has generally been restricted to specific classes of generalized functions ([5], p. 174), (Section I.5.5 in ref. [6]), (Section 8.3 in ref. [7]) and (Section 2.9 in ref. [8]). In addition, the existence of multiple definitions of fractional derivatives often creates difficulties for engineers seeking practical modelling tools. Although several attempts have been made to unify these definitions, such efforts remain only partially successful [4]. Consequently, it has been argued that formulations based on distributions are essential for maintaining structural consistency and backward compatibility with classical calculus [9]. Related distributional representations for other special functions can be found in refs. [10,11,12,13]. More recently, fractional derivatives of the delta function have been studied and implemented in ref. [14]. To the best of our knowledge, a distributional representation of the product of the gamma and zeta functions in terms of fractional derivatives of the Dirac delta distribution has not been previously reported in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

Motivated by the above discussion, we study a generalized distributional framework that provides new insights into fractional calculus. In particular, we develop a novel representation of the product of the Euler gamma and Riemann zeta functions in the form of a fractional Taylor series involving the Dirac delta distribution. By employing fractional operators, we demonstrate how standard functions can be represented within a distributional framework, thereby linking special functions with distribution theory. This formulation enables certain fractional differential equations to be extended from the half-real line $\nu >0$ to the entire real line $\nu \in \mathbb{R}$. Furthermore, generalized (distributional) solutions of the corresponding differential equations are obtained that cannot be derived using classical integer-order derivatives alone, highlighting the essential role of fractional differential operators. The convergence properties of the proposed representation are analyzed, and its Laplace transform is computed explicitly to facilitate the formulation and solution of more general fractional integral and differential equations. Unlike classical approaches that restrict solutions to positive real domains, the present framework enables their extension to the space of generalized functions. Moreover, while many existing fractional models assume regular kernels, the proposed approach introduces a distributional representation of the Gamma–Zeta operator through fractional derivatives of the Dirac delta distribution. This formulation allows the analytical treatment of fractional differential equations with nonremovable singular kernels, which remain inaccessible within classical fractional calculus. The main contributions of this work are summarized as follows:

• A new fractional Taylor-type representation of the product of Euler gamma and Riemann zeta functions is derived using fractional derivatives of the Dirac delta distribution.
• New identities for fractional integrals and derivatives involving multiple Erdélyi–Kober operators are established.
• A distributional framework for fractional differential equations with singular kernels is formulated and solved analytically.
• The approach extends the domain of certain fractional differential equations from $\nu >0$ to $\nu \in \mathbb{R}$ and provides generalized solutions that cannot be obtained using classical integer-order derivatives.

The remainder of the article is organized as follows. Section “Preliminaries” introduces the necessary preliminaries and definitions related to fractional transforms and special functions. Our main results are presented in Section 2. Section 2.1 derives a distributional solution of a fractional differential equation with singular coefficients using the proposed fractional Taylor representation of the Riemann zeta function. Section 2.2 presents new fractional formulas involving the zeta function and is divided into two subsections addressing fractional integrals and fractional derivatives. Section 3 provides additional illustrative examples. Finally, Section 4 concludes the paper and outlines potential directions for future research.

Preliminaries

In this section, we review the necessary preliminaries to present this work. We use some notations throughout, such as $\mathfrak{R}$ is the real part of any complex number, while the set of complex and real numbers is symbolized as $\mathbb{R}$ and $\mathbb{C},$ respectively. Similarly, ${\mathbb{Z}}_0^{-}$ designates the set of negative integers containing 0, while ${\mathbb{R}}^{+}$ contains all positive real numbers.

The Dirac delta function, often known as the $\delta$ (singular) function, is a generalized function [3,6]. The density of an idealized point mass or point charge is physically represented by the delta function, which is frequently conceptualized as an infinitely high, infinitely thin spike at the origin with total area one under the spike. Paul Dirac, a theoretical physicist, first proposed it in 1920. It is frequently called the unit impulse symbol (or function) in the context of signal processing. This function cannot be appropriately specified within the context of classical function theory since it deviates from the integral theory in the Lebesgue sense [3,6]. The fact that differentiation is a continuous operation for the delta function is a key component of this study. Consequently, a number of limiting processes, including infinite summation and integration, commute with distributional differentiation [3,6]. This contrasts with classical analysis, which requires an additional argument to support the inversion of order or prohibits the interchangeability of such operations. Therefore, distributions, also known as generalized functions, present a particular dual space corresponding to a particular test function space. Gelfand and Shilov ([6], Volume 1) provided a thorough treatment of generalized functions. Commonly used test function spaces include the set of smooth functions with compact support $\left(\mathcal{D}\right)$ and the set of quickly decreasing smooth functions ($\mathcal{S}$). Each of these spaces has a matching dual space $\left({\mathcal{D}}^{\prime }\mathrm{a}\mathrm{n}\mathrm{d} {\mathcal{S}}^{\prime }\right)$. $\mathcal{D}$ and ${\mathcal{D}}^{\prime }$ are not Fourier transform invariants, although $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ are. The Fourier transforms converts elements of ${\mathcal{D}}^{\prime }$ to the elements of ${\mathcal{Z}}^{\prime }$, where $\mathcal{Z}$ is a collection of analytic entire test functions $\mathrm{\chi }\left(\mathrm{\omega }\right)$ whose Fourier transforms [3,6] correspond to $\mathcal{D}$. For $\mathrm{\omega }\in \mathbb{C},$ the delta function $\mathrm{\delta }\left(\mathrm{\omega }+\mathrm{b}\right)\in$ ${\mathcal{Z}}^{\prime }$ has the following Taylor series expansion ([3], p. 204) and ([6], Volume 1, p. 259).

$$\mathrm{\delta }\left(\mathrm{\omega }+\mathrm{b}\right)=\sum _{\mathrm{i}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{b}}^{\mathrm{i}}}{\mathrm{i}!}}{\mathrm{\delta }}^{\left(\mathrm{i}\right)}\left(\mathrm{\omega }\right).$$

(1)

Since the fractional version of this series will also be used, we examine some important research on the fractional Taylor series, including Riemann’s formal study of the generalized Taylor series [15]. Then, Hardy [16] verified the evidence of the validity of the Riemann work [15]. Based upon refs. [15,16], Osler [17] has advanced the study of a generalized fractional Taylor series and the generating functions of several special functions. Osler [18] computed Fourier transformations using an integral correspondent of the Taylor series. His contributions inspired other eminent scholars to advance the field of fractional calculus. Since Osler’s work [17,18] broadens the scope for defining and analyzing fractional operators and offers a more cohesive theoretical framework to explore composed functionals [19], we consider the following sum over the general term with $(\gamma =0)$ mentioned in the abstract of ref. [17].

$$\mathrm{\delta }\left(\mathrm{\omega }+\mathrm{b}\right)=\sum _{\mathrm{i}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{b}}^{\mathrm{i}\mathrm{\alpha }}}{\mathrm{\Gamma }(\mathrm{i}\mathrm{\alpha }+1)}}{\mathrm{\delta }}^{\left(\mathrm{i}\mathrm{\alpha }\right)}\left(\mathrm{\omega }\right); 0<\mathrm{\alpha }\le 1.$$

(2)

Likewise, for $\mathrm{\chi }\left(\mathrm{\omega }\right)\in \mathcal{Z}$ the delta function’s ordinary derivatives [3,4].

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{\delta }}^{\left(\mathrm{i}\right)}\left(\mathrm{\omega }\right)\mathrm{\chi }\left(\mathrm{\omega }\right)\mathrm{d}\mathrm{\omega }=\langle {\mathrm{\delta }}^{\left(\mathrm{i}\right)}\left(\mathrm{\omega }\right),\mathrm{\chi }\left(\mathrm{\omega }\right)\rangle ={\left(-1\right)}^{\mathrm{i}}{\mathrm{\chi }}^{\left(\mathrm{i}\right)}\left(0\right),$$

(3)

have been extended to the fractional derivatives in ref. [14]

$$\langle {\mathrm{\delta }}^{\left(\mathrm{i}\mathrm{\alpha }\right)}\left(\mathrm{\omega }\right),\mathrm{\chi }\left(\mathrm{\omega }\right)\rangle ={\left(-1\right)}^{\mathrm{i}\mathrm{\alpha }}{\mathrm{\chi }}^{\left(\mathrm{i}\mathrm{\alpha }\right)}\left(0\right).$$

(4)

However, according to the given function and the type of the derivatives, the limits of integration in (3) and (4) can be different each time. For further details, the interested reader is referred to ref. [14] as well as refs. [20,21,22] and references therein. We know that the delta function contradicts the integral theory in terms of the Lebesgue sense and cannot be properly defined within the framework of classical function theory. Therefore, the new representation cannot be viewed in the classical sense, but using the delta function [3,6] property stated in Equations (3) and (4). It was proved in ref. [22] that the Riemann–Liouville (R–L) and Caputo (C) versions of fractional derivatives of the delta function are equal. Therefore, we do not specify R–L or C in the above Equation (4) and achieve unification.

The gamma function defined by [23],

$$\mathrm{\Gamma }\left(\mathrm{\omega }\right)={\int }_0^{\mathrm{\infty }}{\mathrm{t}}^{\mathrm{\omega }-1}{\mathrm{e}}^{-\mathrm{t}}\mathrm{d}\mathrm{t},\hspace{1em}\hspace{1em}\left(\mathrm{\omega }\in \mathbb{C};\mathfrak{R}\left(\mathrm{\omega }\right)>0\right),$$

(5)

is an extension of the factorial and is, therefore, used as an essential special function.

The Riemann zeta function has the subsequent representation [24]

$$\zeta \left(\mathrm{\omega }\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathrm{\omega }\right)}}{\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{t}}^{\mathrm{\omega }-1}}{{\mathrm{e}}^{\mathrm{t}}-1}}\mathrm{d}\mathrm{t},\hspace{1em}\hspace{1em}(\mathrm{\omega }\in \mathbb{C};\mathfrak{R}\left(\mathrm{\omega }\right)>1).$$

(6)

Actually, (6) is achieved using (5) by following the steps given in ref. ([24], p. 18). Next, we study a class of special functions because we need them to define the multiple Erdélyi–Kober (m-E–K) fractional integral operators used in this research. The kernel of these fractional operators is the Fox H-function. Therefore, let us start by defining the Mittag-Leffler function [25], which is an ordinary replacement for the exponential function in fractional order. It is defined as

$${\mathrm{E}}_{\mathrm{\alpha }}\left(\mathrm{\omega }\right)=\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{\omega }}^{\mathrm{p}}}{\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{p}+1\right)}},\mathrm{\alpha }\in \mathbb{C},\mathfrak{R}\left(\mathrm{\alpha }\right)>0.$$

(7)

It has several important generalizations, such as

$${\mathrm{E}}_{\mathrm{\alpha },\mathrm{\beta }}\left(\mathrm{\omega }\right)=\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{\omega }}^{\mathrm{p}}}{\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{p}+\mathrm{\beta }\right)}},\mathrm{\alpha },\mathrm{\beta }\mathrm{ϵ}\mathbb{C},\mathfrak{R}\left(\mathrm{\alpha }\right)>0$$

(8)

and the famous “Prabhakar function” is also a three-parameter generalization of the Mittag-Leffler function given as [26,27]

$${\mathrm{E}}_{\mathrm{\alpha },\mathrm{\beta }}^{\mathrm{\gamma }}\left(\mathrm{\omega }\right)=\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(\mathrm{\gamma }\right)}_{\mathrm{p}}{\mathrm{\omega }}^{\mathrm{p}}}{\mathrm{p}!\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{p}+\mathrm{\beta }\right)}},\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }\mathrm{ϵ}\mathbb{C},\mathfrak{R}\left(\mathrm{\alpha }\right)>0,$$

(9)

where ${\left(\mathrm{\gamma }\right)}_{\mathrm{p}}$ denotes Pochhammer symbols

$${\left(\mathrm{\gamma }\right)}_{\mathrm{p}}={\displaystyle \frac{\mathrm{\Gamma }\left(\mathrm{\gamma }+\mathrm{p}\right)}{\mathrm{\Gamma }\left(\mathrm{\gamma }\right)}}=\{\begin{array}{c}1 (p=0,)\\ \gamma \left(\mathrm{\gamma }+1\right)\dots \left(\mathrm{\gamma }+\mathrm{n}-1\right) \left(\mathrm{p}\in \mathbb{C}\setminus \left\{0\right\};\mathrm{p}=\mathrm{n}\in \mathbb{N};\mathrm{\gamma }\in \mathbb{C}\right).\end{array}$$

(10)

Mittag-Leffler function also appears in the closed form of some results proved in this research article. Next, considering a suitable Mellin–Barnes category contour $\mathcal{L}$ that splits up the singular points of ${\left\{\mathrm{\Gamma }\left(\begin{array}{c}{1-\mathrm{a}}_{\mathrm{j}}\end{array}-{\mathrm{A}}_{\mathrm{j}}\mathfrak{s}\right)\right\}}_{\mathrm{j}=1}^{\mathrm{n}}$ and ${\left\{\mathrm{\Gamma }\left(\begin{array}{c}{\mathrm{b}}_{\mathrm{j}}\end{array}+{\mathrm{B}}_{\mathrm{j}}\mathfrak{s}\right)\right\}}_{\mathrm{j}=1}^{\mathrm{m}}$, we define the Fox $\mathrm{H}$-function [28] as follows:

$$\begin{array}{c}{\mathrm{H}}_{\mathrm{p},\mathrm{q}}^{\mathrm{m},\mathrm{n}}\left(\mathrm{\omega }\right)={\mathrm{H}}_{\mathrm{p},\mathrm{q}}^{\mathrm{m},\mathrm{n}}\left[\mathrm{\omega }|\begin{array}{c}\left({\mathrm{a}}_{\mathrm{i}},{\mathrm{A}}_{\mathrm{i}}\right)\\ \left({\mathrm{b}}_{\mathrm{j}},{\mathrm{B}}_{\mathrm{j}}\right)\end{array}\right]={\mathrm{H}}_{\mathrm{p},\mathrm{q}}^{\mathrm{m},\mathrm{n}}\left[\mathrm{z}|\begin{array}{cc}\left({\mathrm{a}}_1,{\mathrm{A}}_1\right),\dots ,& \left({\mathrm{a}}_{\mathrm{i}},{\mathrm{A}}_{\mathrm{i}}\right)\\ \left({\mathrm{b}}_1,{\mathrm{B}}_1\right),\dots ,& \left({\mathrm{b}}_{\mathrm{j}},{\mathrm{B}}_{\mathrm{j}}\right)\end{array}\right]\\ ={\displaystyle \frac{1}{2\mathrm{\pi }\mathrm{i}}}{\int }_{\mathcal{L}}{\displaystyle \frac{\prod _{\mathrm{j}=1}^{\mathrm{m}}\mathrm{\Gamma }\left(\begin{array}{c}{\mathrm{b}}_{\mathrm{j}}\end{array}+{\mathrm{B}}_{\mathrm{j}}\mathfrak{s}\right)\prod _{\mathrm{i}=1}^{\mathrm{n}}\mathrm{\Gamma }\left(\begin{array}{c}{1-\mathrm{a}}_{\mathrm{j}}\end{array}-{\mathrm{A}}_{\mathrm{j}}\mathfrak{s}\right)}{\prod _{\mathrm{j}=\mathrm{m}+1}^{\mathrm{q}}\mathrm{\Gamma }\left(\begin{array}{c}{1-\mathrm{b}}_{\mathrm{j}}\end{array}-{\mathrm{B}}_{\mathrm{j}}\mathfrak{s}\right)\prod _{\mathrm{i}=\mathrm{n}+1}^{\mathrm{p}}\mathrm{\Gamma }\left(\begin{array}{c}{\mathrm{a}}_{\mathrm{j}}\end{array}+{\mathrm{A}}_{\mathrm{j}}\mathfrak{s}\right)}}{\mathrm{\omega }}^{-\mathfrak{s}}\mathrm{d}\mathfrak{s},\\ (1\le \mathrm{m}\le \mathrm{q};1\le \mathrm{n}\le \mathrm{p},{\mathrm{A}}_{\mathrm{i}}>0;{\mathrm{B}}_{\mathrm{j}}>0;{\mathrm{a}}_{\mathrm{i}}\in \mathbb{C};{\mathrm{b}}_{\mathrm{j}}\in \mathbb{C}).\end{array}$$

(11)

For the special consideration of parameters ${\mathrm{A}}_{\mathrm{p}}={\mathrm{B}}_{\mathrm{q}}=1,$ $\mathrm{H}$-function turns out to be a Meijer $\mathrm{G}$-function [28]

$${\mathrm{H}}_{\mathrm{p},\mathrm{q}}^{\mathrm{m},\mathrm{n}}\left[\mathrm{\omega }|\begin{array}{cc}\left({\mathrm{a}}_1,1\right),\dots ,& \left({\mathrm{a}}_{\mathrm{i}},1\right)\\ \left({\mathrm{b}}_1,1\right),\dots ,& \left({\mathrm{b}}_{\mathrm{j}},1\right)\end{array}\right]={\mathrm{G}}_{\mathrm{p},\mathrm{q}}^{\mathrm{m},\mathrm{n}}\left[\mathrm{\omega }\left|\begin{array}{cc}{\mathrm{a}}_1\dots ,& {\mathrm{a}}_{\mathrm{i}}\\ {\mathrm{b}}_1,\dots ,& {\mathrm{b}}_{\mathrm{j}}\end{array}\right|\right].$$

(12)

Here and what follows the Fox–Wright function ${}_{\mathrm{p}}\mathrm{\Psi }_{\mathrm{q}}$ has the ensuing relationship with Fox $\mathrm{H}$-function [28]

$$\begin{array}{cc}\hfill {}_{\mathrm{p}}\mathrm{\Psi }_{\mathrm{q}}\left[\begin{array}{c}\left({\mathrm{a}}_{\mathrm{i}},{\mathrm{A}}_{\mathrm{i}}\right)\\ {(\mathrm{b}}_{\mathrm{j}},{\mathrm{B}}_{\mathrm{j}})\end{array};\mathrm{\omega }\right]& ={\displaystyle \sum _{\mathrm{m}=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\displaystyle {\prod }_{\mathrm{l}=1}^{\mathrm{p}}}\mathrm{\Gamma }\left({\mathrm{a}}_{\mathrm{i}}+{\mathrm{A}}_{\mathrm{i}}\mathrm{m}\right)}{{\displaystyle {\prod }_{\mathrm{l}=1}^{\mathrm{q}}}\mathrm{\Gamma }{(\mathrm{b}}_{\mathrm{j}}+{\mathrm{B}}_{\mathrm{j}}\mathrm{m})}}{\displaystyle \frac{{\mathrm{\omega }}^{\mathrm{m}}}{\mathrm{m}!}}\hfill \\ & ={\mathrm{H}}_{\mathrm{p},\mathrm{q}+1}^{1,\mathrm{p}}\left[-\mathrm{\omega }|\begin{array}{cc}\left(1-{\mathrm{a}}_1,{\mathrm{A}}_1\right),\dots ,& \left(1-{\mathrm{a}}_{\mathrm{i}},{\mathrm{A}}_{\mathrm{i}}\right)\\ \left(0,1\right),\left({1-\mathrm{b}}_1,{\mathrm{B}}_1\right),\dots ,& {(1-\mathrm{b}}_{\mathrm{j}},{\mathrm{B}}_{\mathrm{j}})\end{array}\right]\hfill \\ \hfill ({\mathrm{a}}_{\mathrm{i}}\mathrm{ϵ}{\mathbb{R}}^{+}\mathrm{i}=1,& \dots .\mathrm{p});{\mathrm{B}}_{\mathrm{j}}\mathrm{ϵ}{\mathbb{R}}^{+}(\mathrm{j}=1,\dots .\mathrm{q});1+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{q}}}{\mathrm{B}}_{\mathrm{i}}-{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{p}}}{\mathrm{A}}_{\mathrm{j}}>0)\hfill \end{array}$$

(13)

and for $\left({\mathrm{a}}_{\mathrm{i}}>0;{\mathrm{b}}_{\mathrm{j}}\notin {\mathbb{Z}}_0^{-}\right)$, it also comprises the hypergeometric ${}_{\mathrm{p}}\mathrm{F}_{\mathrm{q}}$ and Meijer $G$-functions [28] as follows:

$$\begin{array}{cc}\hfill {}_{\mathrm{p}}\mathrm{\Psi }_{\mathrm{q}}\left[\begin{array}{c}\left({\mathrm{a}}_{\mathrm{i}},1\right)\\ {(\mathrm{b}}_{\mathrm{j}},1)\end{array};\mathrm{\omega }\right]& {=\mathrm{G}}_{\mathrm{p},\mathrm{q}+1}^{1,\mathrm{p}}\left[-\mathrm{\omega }|\begin{array}{cc}\left({1-\mathrm{a}}_1,1\right),\dots ,& \left(1-{\mathrm{a}}_{\mathrm{i}},1\right)\\ 0,\left(1-{\mathrm{b}}_1,1\right),\dots ,& \left(1-{\mathrm{b}}_{\mathrm{j}},1\right)\end{array}\right]\hfill \\ & ={}_{\mathrm{p}}\mathrm{F}_{\mathrm{q}}\left[\begin{array}{c}{\mathrm{a}}_{\mathrm{i}}\\ {\mathrm{b}}_{\mathrm{j}}\end{array};\mathrm{\omega }\right].{\displaystyle \frac{\mathrm{\Gamma }\left({\mathrm{a}}_1\right)\dots \mathrm{\Gamma }\left({\mathrm{a}}_{\mathrm{i}}\right)}{\mathrm{\Gamma }\left({\mathrm{b}}_1\right)\dots \mathrm{\Gamma }\left({\mathrm{b}}_{\mathrm{j}}\right)}}.\hfill \end{array}$$

(14)

Moreover, the above functions are also related with Mittag-Leffler functions as follows [28]:

$$\begin{gathered} {\mathrm{E}}_{\mathrm{\alpha },\mathrm{\beta }}^{\mathrm{\gamma }}\left(\mathrm{\omega }\right)=\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(\mathrm{\gamma }\right)}_{\mathrm{p}}{\mathrm{\omega }}^{\mathrm{p}}}{\mathrm{p}!\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{p}+\mathrm{\beta }\right)}}={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathrm{\gamma }\right)}}{}_1\mathrm{\Psi }_1\left[\begin{array}{c}\left(\mathrm{\gamma },1\right)\\ \left(\mathrm{\beta },\mathrm{\alpha }\right)\end{array};\mathrm{\omega }\right]={\mathrm{H}}_{\mathrm{1,2}}^{\mathrm{1,1}}\left[-\mathrm{\omega }|\begin{array}{c}\left(1-\mathrm{\gamma },1\right)\\ \left(\mathrm{0,1}\right),\left(1-\mathrm{\beta },\mathrm{\alpha }\right)\end{array}\right]; \\ {\mathrm{E}}_{\mathrm{\alpha },\mathrm{\beta }}^1\left(\mathrm{\omega }\right)={\mathrm{E}}_{\mathrm{\alpha },\mathrm{\beta }}\left(\mathrm{\omega }\right)=\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{\omega }}^{\mathrm{p}}}{\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{p}+\mathrm{\beta }\right)}}={}_1\mathrm{\Psi }_1\left[\begin{array}{c}\left(1,1\right)\\ \left(\mathrm{\beta },\mathrm{\alpha }\right)\end{array};\mathrm{\omega }\right]={\mathrm{H}}_{\mathrm{1,2}}^{\mathrm{1,1}}\left[-\mathrm{\omega }|\begin{array}{c}\left(\mathrm{0,1}\right)\\ \left(\mathrm{0,1}\right),\left(1-\mathrm{\beta },\mathrm{\alpha }\right)\end{array}\right]; \\ {\mathrm{E}}_{\mathrm{\alpha },1}^1\left(\mathrm{\omega }\right)={\mathrm{E}}_{\mathrm{\alpha }}\left(\mathrm{\omega }\right)=\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{\omega }}^{\mathrm{p}}}{\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{p}+1\right)}}={}_1\mathrm{\Psi }_1\left[\begin{array}{c}\left(1,1\right)\\ \left(1,\mathrm{\alpha }\right)\end{array};\mathrm{\omega }\right]={\mathrm{H}}_{\mathrm{1,2}}^{\mathrm{1,1}}\left[-\mathrm{\omega }|\begin{array}{c}\left(\mathrm{0,1}\right)\\ \left(\mathrm{0,1}\right),\left(0,\mathrm{\alpha }\right)\end{array}\right]. \end{gathered}$$

(15)

These relations are important to analyze the closed form of our results in relation to Fox–Wright and Fox H-functions. Furthermore, there is a strong connection between the special functions and the fractional calculus; likewise, the multiple Erdélyi–Kober (m-E–K) fractional integral operators (Equation (18) in ref. [29], p. 8) are defined by the H-function in their kernel

$${\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\mathrm{f}\left(\mathrm{z}\right)=\{\begin{array}{c}{{\int }_0^{1}f\left(z\sigma \right) H}_{m,m}^{m,0}\left[\sigma |\begin{array}{c}{\left({\gamma }_k+{\mathrm{\nu }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\beta }_k}},{\displaystyle \frac{1}{{\beta }_k}}\right)}_1^m\\ {\left({\gamma }_k+1-{\displaystyle \frac{1}{{\beta }_k}},{\displaystyle \frac{1}{{\beta }_k}}\right)}_1^m\end{array} \right]d\sigma ;{\displaystyle \sum _k}{\mathrm{\nu }}_{\mathrm{k}}>0 \\ {{=z}^{-1}{\int }_0^{z}f\left(\xi \right) H}_{m,m}^{m,0}\left[{\displaystyle \frac{\xi }{z}}|\begin{array}{c}{\left({\gamma }_k+{\mathrm{\nu }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\beta }_k}},{\displaystyle \frac{1}{{\beta }_k}}\right)}_1^m\\ {\left({\gamma }_k+1-{\displaystyle \frac{1}{{\beta }_k}},{\displaystyle \frac{1}{{\beta }_k}}\right)}_1^m\end{array} \right]d\xi ;{\displaystyle \sum _k}{\mathrm{\nu }}_{\mathrm{k}}>0 \\ f\left(z\right);{\delta }_k=1\end{array}$$

(16)

We use substitution $\sigma ={\displaystyle \frac{\xi }{z}}$ in the first row of (16) to get second equivalent form. The fractional multi-order of integration is denoted by ${\mathrm{\nu }}_{\mathrm{k}}$, ${\mathrm{\gamma }}_{\mathrm{k}}\prime \mathrm{s}$ are used to denote weights, while some supplementary parameters are denoted by ${\mathrm{\beta }}_{\mathrm{k}}>0.$ Since ${\mathrm{H}}_{\mathrm{m},\mathrm{m}}^{\mathrm{m},0}$ becomes zero for $\left|\mathrm{\sigma }\right|>1$, one can consider $\sigma \in \left(0,\mathrm{\infty }\right).$

The corresponding fractional derivative is defined as ([29], p. 9) (see also ref. [30])

$${\mathrm{D}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{{\left({\mathrm{\gamma }}_{\mathrm{k}}\right)}_1^{\mathrm{m}},\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left(\mathrm{f}\left(\mathrm{z}\right)\right):={\mathrm{D}}_{\mathrm{\eta }}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\nu }}_{\mathrm{k}}\right),\left(\mathrm{\eta }\mathrm{k}-{\mathrm{\nu }}_{\mathrm{k}}\right)}\mathrm{f}\left(\mathrm{z}\right)={\mathrm{D}}_{\mathrm{\eta }}{\int }_0^1\mathrm{f}\left(\mathrm{z}\mathrm{\sigma }\right){\mathrm{H}}_{\mathrm{m},\mathrm{m}}^{\mathrm{m},0}\left[\mathrm{\sigma }|\begin{array}{c}{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\eta }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\\ {\left({\mathrm{\gamma }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\end{array}\right]\mathrm{d}\mathrm{\sigma }$$

(17)

or

$${\mathrm{D}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{{\left({\mathrm{\gamma }}_{\mathrm{k}}\right)}_1^{\mathrm{m}},\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left(\mathrm{f}\left(\mathrm{z}\right)\right):={\mathrm{D}}_{\mathrm{\eta }}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\nu }}_{\mathrm{k}}\right),\left({\mathrm{\eta }}_{\mathrm{k}}-{\mathrm{\nu }}_{\mathrm{k}}\right)}\mathrm{f}\left(\mathrm{z}\right)={\mathrm{D}}_{\mathrm{\eta }}{\int }_0^z\mathrm{f}\left(\xi \right){\mathrm{H}}_{\mathrm{m},\mathrm{m}}^{\mathrm{m},0}\left[{\displaystyle \frac{\xi }{z}}|\begin{array}{c}{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\eta }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\\ {\left({\mathrm{\gamma }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\end{array}\right]\mathrm{d}\xi$$

(18)

where ${\mathrm{D}}_{\mathrm{\eta }}$ is a polynomial in variable $z{\displaystyle \frac{d}{dz}}$ of degree ${\mathrm{\eta }}_1+...+{\mathrm{\eta }}_{\mathrm{m}}$, given by

$${\mathrm{D}}_{\mathrm{\eta }}={\mathrm{P}}_{\mathrm{\eta }}\left(z{\displaystyle \frac{d}{dz}}\right) {=\prod }_{\mathrm{r}=1}^{\mathrm{m}}{\prod }_{\mathrm{j}=1}^{{\mathrm{\eta }}_{\mathrm{r}}}{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{r}}}}\mathrm{z}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{z}}}+{\mathrm{\gamma }}_{\mathrm{r}}+\mathrm{j};\hspace{1em}{\mathrm{\eta }}_{\mathrm{k}}=\{\begin{array}{c}\left(\right[{\mathrm{\nu }}_{\mathrm{k}}]+1;{\mathrm{\nu }}_{\mathrm{k}}\notin \mathbb{Z}\\ {\mathrm{\nu }}_{\mathrm{k}};{\mathrm{\nu }}_{\mathrm{k}}\in \mathbb{Z}\end{array}.$$

(19)

and the C version is defined as follows [30]:

$${{}_{\mathrm{*}}\mathrm{D}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{{\left({\mathrm{\gamma }}_{\mathrm{k}}\right)}_1^{\mathrm{m}},\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left(\mathrm{f}\left(\mathrm{z}\right)\right):={\mathrm{D}}_{\mathrm{\eta }}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\nu }}_{\mathrm{k}}\right),\left(\mathrm{\eta }\mathrm{k}-{\mathrm{\nu }}_{\mathrm{k}}\right)}\mathrm{f}\left(\mathrm{z}\right)={\int }_0^1{\mathrm{D}}_{\mathrm{\eta }}\mathrm{f}\left(\mathrm{z}\mathrm{\sigma }\right){\mathrm{H}}_{\mathrm{m},\mathrm{m}}^{\mathrm{m},0}\left[\mathrm{\sigma }|\begin{array}{c}{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\eta }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\\ {\left({\mathrm{\gamma }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\end{array}\right]\mathrm{d}\mathrm{\sigma }$$

(20)

or

$${{}_{\mathrm{*}}\mathrm{D}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{{\left({\mathrm{\gamma }}_{\mathrm{k}}\right)}_1^{\mathrm{m}},\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left(\mathrm{f}\left(\mathrm{z}\right)\right):={\mathrm{D}}_{\mathrm{\eta }}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\nu }}_{\mathrm{k}}\right),\left({\mathrm{\eta }}_{\mathrm{k}}-{\mathrm{\nu }}_{\mathrm{k}}\right)}\mathrm{f}\left(\mathrm{z}\right)={\int }_0^z{\mathrm{D}}_{\mathrm{\eta }}\mathrm{f}\left(\xi \right){\mathrm{H}}_{\mathrm{m},\mathrm{m}}^{\mathrm{m},0}\left[{\displaystyle \frac{\xi }{z}}|\begin{array}{c}{\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\eta }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\\ {\left({\mathrm{\gamma }}_{\mathrm{k}}+1-{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\end{array}\right]\mathrm{d}\xi$$

(21)

In this article, we use C derivative only in Case 3 of Theorem 2; otherwise, all the results and operators are considered in the R–L sense. Limits of these integrals are considered from 0 to z. We can use Equations (16), (18) and (21) in conjunction with the following

Table 1. Imperative special cases of m-E–K operators [29,30].

$\mathbf{Special} \mathbf{Cases} \mathbf{of} {\mathbf{H}}_{\mathbf{m},\mathbf{m}}^{\mathbf{m},0}$	Connection Among the Individual Kernels of the Preceding Fractional Operators
(Equation (31) in ref. [29]) Marichev–Saigo–Maeda (M–S–M) $\left(m=3;{\beta }_1={\beta }_2={\beta }_3=\beta =1\right)$	$\begin{array}{c}{\mathrm{H}}_{\mathrm{3,3}}^{\mathrm{3,0}} \left({\displaystyle \frac{\mathrm{t}}{\mathrm{x}}}\right)={\mathrm{G}}_{\mathrm{3,3}}^{\mathrm{3,0}}\left[{\displaystyle \frac{\mathrm{t}}{\mathrm{x}}}|\begin{array}{c}{{\gamma }_1}^{\prime }+{{\gamma }_2}^{\prime }, \nu -{\gamma }_1,\nu -{\gamma }_2 \\ {{\gamma }_1}^{\prime },{{\gamma }_2}^{\prime }, \nu -{\gamma }_1-{\gamma }_2\end{array} \right]\hfill \\ \hfill ={\displaystyle \frac{{\mathrm{x}}^{-{\gamma }_1}}{\mathrm{\Gamma }\left(\nu \right)}} {\left(\mathrm{x}-\mathrm{t}\right)}^{\mathrm{\delta }-1}{\mathrm{t}}^{-{{\gamma }_1}^{\prime }}{\mathrm{F}}_3\left({\gamma }_1,{{\gamma }_1}^{\prime },{\gamma }_2,{{\gamma }_2}^{\prime },\nu ;1-{\displaystyle \frac{\mathrm{t}}{\mathrm{x}}};1-{\displaystyle \frac{\mathrm{x}}{\mathrm{t}}}\right)\end{array}$
(Equation (30) in ref. [29]) Saigo $\left(\mathrm{m}=2;{\mathrm{\beta }}_1={\mathrm{\beta }}_2=1;\mathrm{\sigma }={\displaystyle \frac{\mathrm{t}}{\mathrm{x}}}\wedge \mathrm{\sigma }={\displaystyle \frac{\mathrm{x}}{\mathrm{t}}}\right)$	$\begin{array}{c}{\mathrm{H}}_{\mathrm{2,2}}^{\mathrm{2,0}}\left[\mathrm{\sigma }|\begin{array}{c}\left({\gamma }_1+{\nu }_1,1\right),\left({\gamma }_2+{\nu }_2,1\right)\\ \left({\gamma }_1,1\right),\left({\gamma }_2,1\right)\end{array} \right]={\mathrm{G}}_{\mathrm{2,2}}^{\mathrm{2,0}}\left[\mathrm{\sigma }|\begin{array}{c}{\gamma }_1+{\nu }_1,{\gamma }_2+{\nu }_2\\ {\gamma }_1,{\gamma }_2\end{array}\right]\hfill \\ \hfill =\mathrm{\beta }{\displaystyle \frac{{\mathrm{\sigma }}^{{\gamma }_2}{\left(1-\mathrm{\sigma }\right)}^{{\nu }_1+{\nu }_2-1}}{\mathrm{\Gamma }\left({\nu }_1+{\nu }_2\right)}}{}_2\mathrm{F}_1\left({\gamma }_2+{\nu }_2-{\gamma }_1,{\nu }_1;{\nu }_1+{\nu }_2;1-\mathrm{\sigma }\right)\end{array}$
([29], p. 10) Erdélyi–Kober (E–K) (m = 1)	${\mathrm{H}}_{\mathrm{1,0}}^{\mathrm{1,1}}\left[\mathrm{\sigma }|\begin{array}{c}\left(\nu ,{\displaystyle \frac{1}{\mathrm{\beta }}}\right)\\ \left(0,{\displaystyle \frac{1}{\mathrm{\beta }}}\right)\end{array}\right]=\mathrm{\beta }{\mathrm{\sigma }}^{\mathrm{\beta }-1}{\mathrm{G}}_{\mathrm{1,0}}^{\mathrm{1,1}}\left[{\mathrm{\sigma }}^{\mathrm{\beta }}|\begin{array}{c}\nu \\ 0\end{array}\right]={\displaystyle \frac{{\mathrm{\beta }\mathrm{\sigma }}^{\mathrm{\beta }-1}{\left(1-{\mathrm{\sigma }}^{\mathrm{\beta }}\right)}^{\nu -1}}{\mathrm{\Gamma }\left(\nu \right)}}$
([29], p. 10) Riemann–Liouville (R–L) $(\mathrm{m}=1;\mathrm{\beta }=1;\mathrm{\sigma }={\displaystyle \frac{\mathrm{t}}{x}}\wedge \mathrm{\sigma }={\displaystyle \frac{\mathrm{x}}{t}})$	${\mathrm{H}}_{\mathrm{1,0}}^{\mathrm{1,1}}\left[\mathrm{\sigma }|\begin{array}{c}\left(\nu ,1\right)\\ \left(\mathrm{0,1}\right)\end{array}\right]={\mathrm{G}}_{\mathrm{1,0}}^{\mathrm{1,1}}\left[{\displaystyle \frac{\mathrm{t}}{x}}|\begin{array}{c}\nu \\ -\end{array}\right]={\displaystyle \frac{{\left(\mathrm{x}-\mathrm{t}\right)}^{\nu -1}}{\mathrm{\Gamma }\left(\nu \right)}}$

and generate various fractional derivatives and integral transforms. However, for a more detailed discussion of such operators in the R–L and C sense, the interested reader is referred to refs. [29,30].

2. Main Results

Our main findings are listed in this section by dividing it into the subsequent subsections.

2.1. A Distributional Framework Based on Gamma–Zeta Operator

Because of their numerous uses, the generalizations of the Riemann zeta function [31] and the function itself have always been fundamentally important. For example, to explore the complex dimensions of fractal strings, fractal geometry relies heavily on the Riemann zeta function [32]. A function is generally understood to be defined by a series, an integral of a particular variable, or in terms of those functions that we designate “elementary.” It is necessary to consider the function as a separate entity that can be represented by an integral or a series. The importance of this notion increases when one considers the theory of (special) functions. There are multiple representations for every special function, including asymptotic and series integral representations, and in this section, we present and analyze a fractional distributional representation of the Riemann zeta function. This representation is novel for computing the answers of incomputable singular integrals acting as an operator. At regular points such as $\mathrm{\omega }=2$, the classical form (6) can be evaluated directly. The new fractional delta expansion is primarily intended for encoding singularities; its value at regular points is interpreted in the sense of distributions under integration. The classical form (6) diverges at $\mathrm{\omega }=1$, reflecting a singularity that standard calculus cannot handle. In contrast, the new fractional delta expansion in the subsequent theorem represents this singularity as a series of fractional delta derivatives, ${\mathrm{\delta }}^{\left(\mathrm{r}\mathrm{\alpha }\right)}\left(\mathrm{\omega }\right)$, each acting as a “refined impulse” that encodes the singular behaviour at increasingly subtle scales. This transforms an otherwise divergent expression into a well-defined, analytically tractable series, providing both a rigorous mathematical framework and a physically intuitive tool for modelling phenomena in fractional calculus, anomalous diffusion, and quantum systems with memory or nonlocal effects. Since the product of the Euler gamma and Riemann zeta functions appears in this new representation, we name it the Gamma–Zeta operator.

Theorem 1. 

The Gamma–Zeta operator can be signified in the series form as follows:

$${\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right)=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\delta }^{\left(r\alpha \right)}\left(\omega \right); 0<\alpha \le 1.$$

(22)

Proof. 

Mellin transform representation (6) can be rewritten as the Fourier transform representation by using ${\displaystyle \frac{1}{{\mathrm{e}}^{\mathrm{t}}-1}}={\displaystyle \frac{{\mathrm{e}}^{-\mathrm{t}}}{{1-\mathrm{e}}^{-\mathrm{t}}}}$ and substituting $\mathrm{\omega }=\mathrm{\sigma }+\mathrm{\iota }\mathrm{\tau };t=e^y;dt=e^{y}dy;t=0;y=-\mathrm{\infty };\mathrm{t}=\mathrm{\infty };\mathrm{y}=\mathrm{\infty }$ in (6) as follows:

$$\mathrm{\Gamma }\left(\mathrm{\omega }\right)\zeta \left(\mathrm{\omega }\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{e^{\left(\mathrm{\sigma }+\mathrm{\iota }\mathrm{\tau }\right)y-y}{e}^y{\mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{e}}^y)}{1-{\mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{e}}^y)}}\mathrm{d}\mathrm{y},\hspace{1em}\hspace{1em}(\mathfrak{R}(\mathrm{\omega })=\mathrm{\sigma }>1).$$

(23)

Next, expanding the term in (23) as follows:

$${\displaystyle \frac{{\mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{e}}^y)}{1-{\mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{e}}^y)}}=\sum _{n=0}^{\mathrm{\infty }}{\mathrm{e}\mathrm{x}\mathrm{p}(-(\mathrm{n}+1)\mathrm{e}}^y)=\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{{(-1)}^p{\left(\mathrm{n}+1\right)}^p\mathrm{e}}^{py}}{p!}}.$$

(24)

Since uniform convergence of the integral allows us to change the order of summation and integration. Therefore, using (24) by summing over $p$ first in (23) and denoting the Fourier transform by $\mathcal{F},$ we rewrite (23) as

$$\mathrm{\Gamma }\left(\mathrm{\omega }\right)\zeta \left(\mathrm{\omega }\right)=\mathcal{F}\left[\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{{(-1)}^p{\left(\mathrm{n}+1\right)}^p\mathrm{e}}^{(\mathrm{\sigma }+\mathrm{p})y}}{p!}};\mathrm{\tau }\right]=\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{(-1)}^p{\left(\mathrm{n}+1\right)}^p}{p!}}\mathcal{F}\left[{\mathrm{e}}^{(\mathrm{\sigma }+\mathrm{p})y};\mathrm{\tau }\right].$$

(25)

However, using ref. ([3], p. 204) or ref. ([6], Volume 1, p.169) and basic properties of the delta function, we obtain

$$\mathcal{F}\left[{\mathrm{e}}^{(\mathrm{\sigma }+\mathrm{p})y};\mathrm{\tau }\right]=2\pi \delta \left(\tau -\iota \left(\sigma +p\right)\right)=2\pi \delta \left({\displaystyle \frac{1}{\iota }} (\iota \tau -{\iota }^2(\sigma +p)\right)=2\pi \left|\iota \right|\delta \left(\sigma +\iota \tau +p\right)=2\mathrm{\pi }\delta \left(\omega +p\right).$$

Using it in (25) will yield the following

$$\mathrm{\Gamma }\left(\mathrm{\omega }\right)\zeta \left(\mathrm{\omega }\right)=2\pi \sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{(-1)}^p{\left(\mathrm{n}+1\right)}^p}{p!}}\delta \left(\omega +p\right).$$

(26)

First, by summing over $p$, the coefficients $\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{(-1)}^p{\left(\mathrm{n}+1\right)}^p}{p!}}={\displaystyle \frac{1}{e-1}}$ of $\delta \left(\omega +p\right)$ in (26) are independent of the function argument $\omega ;$ therefore, using the fractional Taylor series (2), we obtain

$${\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\zeta }_{\alpha }\left(\mathrm{\omega }\right)=2\pi \sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{(-1)}^p{\left(\mathrm{n}+1\right)}^p}{p!}}\sum _{r=0}^{\infty }{\displaystyle \frac{b^{r\alpha }}{\Gamma (r\alpha +1)}}{\delta }^{\left(r\alpha \right)}\left(\omega \right).$$

(27)

It leads to the required result, which is a distributional representation defined for suitably chosen functions using (4). However, it is obvious that by reversing the steps we can obtain (6) from (27). □

Remark 1. 

As mentioned before in Section “Preliminaries”, we recall using the same definitions of the R–L and C versions of the fractional derivatives as given in ref. [30]. However, the following theorem proves that the R–L and C versions of the new representation are the same.

Theorem 2. 

For $\mathrm{t}>0$, $\mathrm{\nu }\notin \mathbb{N},\mathrm{\tau }\in [0,\mathrm{t})$ prove that

$${{{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right).$$

(28)

Proof. 

We consider three cases according to the value of $\mathrm{\nu }\notin \mathbb{N}.$

• Case 1: When $\mathit{\nu }$ < 0, we use (Equation (1) in ref. [30]) for the new representation (22)

$$\begin{gathered} {{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={{{}_0{}^R\mathrm{I}}_{\mathrm{t}}^{-\nu }\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\tau }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\tau }\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(-\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{-\nu -1}{\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\tau }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau } \\ {{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(-\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{-\nu -1}2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\delta }^{\left(r\alpha \right)}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }. \end{gathered}$$

(29)

The symbol $0^{-}$ used as the lower integration limit in Equation (29) refers to a limiting process in which the integration interval extends from a vanishingly small negative value to infinity. This approach ensures that all singular contributions arising from the derivatives of the delta function ${\displaystyle \frac{{\delta }^{\left(r\alpha \right)}\left(\mathrm{\tau }-0\right)}{d\tau }}$ are fully accounted for within the integral formulation. Since distributional differentiation commutes with various limiting processes such as infinite summation and integration [3,6], therefore

$${}_0{}^R\mathrm{D}_{\mathrm{t}}^{\nu }{\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\Gamma }\left(-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{-\nu -1}{\delta }^{\left(r\alpha \right)}\left(\tau \right)\mathrm{d}\mathrm{\tau }.$$

Next, using (4), we get

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\Gamma }\left(-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\displaystyle \frac{{\left(-1\right)}^{r\alpha }{\mathrm{d}}^{\left(r\alpha \right)}}{d{\tau }^{r\alpha }}}{\left[{\left(t-\tau \right)}^{-\nu -1}\right]}_{\tau =0}.$$

(30)

Here and what follows (Equation (3) in ref. [30])

$${{\displaystyle \frac{d^{r\alpha }}{d{\mathrm{\tau }}^{r\alpha }}}\left(t-\tau \right)}^{-\nu -1}={\displaystyle \frac{{\left(-1\right)}^{r\alpha }\Gamma \left(-\nu \right){\left(t-\tau \right)}^{\nu -1-r\alpha }}{\Gamma \left(-\nu -r\alpha \right)}},$$

(31)

and using it in (30), we get

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }\mathrm{\Gamma }\left(-\nu \right)}{\mathrm{\Gamma }\left(-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\displaystyle \frac{{\left(t\right)}^{-\nu -1-r\alpha }}{\Gamma \left(-\nu -r\alpha \right)}}.$$

After some further modifications, we obtain the final result as follows

$${{{}_0{}^{R}D}_t^{\nu }\Gamma }_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={{{}_0{}^{R}I}_t^{-\nu }\Gamma }_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{t^{\nu +1}}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(\frac{p}{t}\right)}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)\Gamma \left(-\nu -r\alpha \right)}}.$$

(32)

• Case 2: When $\mathit{m}-\mathbf{1}<\mathit{\nu }\le \mathit{m};\mathit{m}\in \mathbb{N},$ we use (Equation (2) in ref. [30]) for the new representation (22)

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{d^m}{d{t}^m}}{{}_0{}^R\mathrm{I}}_{\mathrm{t}}^{m-\nu }{\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(m-\nu \right)}}{\displaystyle \frac{d^m}{d{t}^m}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{m-\nu -1}{\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\tau }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau },$$

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{1}{\Gamma \left(m-\nu \right)}}{\displaystyle \frac{d^m}{d{t}^m}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{m-\nu -1}2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\delta }^{\left(r\alpha \right)}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }.$$

Since distributional differentiation commutes with various limiting processes such as infinite summation and integration [3,6], it follows that

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\Gamma }\left(m-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\displaystyle \frac{d^m}{d{t}^m}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{m-\nu -1}{\delta }^{\left(r\alpha \right)}\left(\tau \right)\mathrm{d}\mathrm{\tau }.$$

Next, using (4), we obtain

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\Gamma }\left(m-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\displaystyle \frac{d^m}{d{t}^m}}\left[{\displaystyle \frac{{\left(-1\right)}^{r\alpha }{\mathrm{d}}^{\left(r\alpha \right)}}{d{\tau }^{r\alpha }}}{\left[{\left(t-\tau \right)}^{m-\nu -1}\right]}_{\tau =0}\right],$$

which can be rewritten as follows using (31)

$${{{}_0{}^R\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }\mathrm{\Gamma }\left(m-\nu \right)}{\mathrm{\Gamma }\left(m-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(p\right)}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\displaystyle \frac{{\frac{d^m}{d{t}^m}\left(t\right)}^{m-\nu -1-r\alpha }}{\Gamma \left(m-\nu -r\alpha \right)}}.$$

Again, differentiating it $m$ times in the classical sense, and deleting each time a factor, we get the final form

$${{{}_0{}^{R}D}_t^{\nu }\Gamma }_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={{{}_0{}^{R}I}_t^{-\nu }\Gamma }_{\alpha }\left(\mathrm{\tau }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\tau }\right)={\displaystyle \frac{2\mathrm{\pi }}{t^{\nu +1}}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(\frac{p}{t}\right)}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)\Gamma \left(-\nu -r\alpha \right)}}.$$

(33)

• Case 3: When  $\mathit{m}-\mathbf{1}<\mathit{\nu }\le \mathit{m}; \mathit{m}\in \mathbb{N}$⁺, we use (Equation (4) in ref. [30]) for the new representation (22)

$$\begin{array}{cc}\hfill {{{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)=& {\displaystyle \frac{d^m}{d{t}^m}}{{}_0{}^C\mathrm{I}}_{\mathrm{t}}^{m-\nu }{\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\tau }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\tau }\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(m-\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{m-\nu -1}{\displaystyle \frac{d^m}{d{\mathrm{\tau }}^m}}{\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\tau }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }\hfill \\ & ={\displaystyle \frac{1}{\Gamma \left(m-\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{m-\nu -1}2\mathrm{\pi }{\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\displaystyle \frac{d^m}{d{\mathrm{\tau }}^m}}{\delta }^{\left(r\alpha \right)}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }.\hfill \end{array}$$

Since distributional differentiation commutes with various limiting processes such as infinite summation and integration [3,6], it follows that

$$\begin{gathered} {{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }{\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\Gamma }\left(m-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{m-\nu -1}{\displaystyle \frac{d^m}{d{\mathrm{\tau }}^m}}{\delta }^{\left(r\alpha \right)}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau } \\ ={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{\Gamma }\left(m-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(p\right)}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\left[{\displaystyle \frac{d^m}{d{\mathrm{\tau }}^m}}\left({{\displaystyle \frac{d^{r\alpha }}{d{\mathrm{\tau }}^{r\alpha }}}\left(t-\tau \right)}^{m-\nu -1}\right)\right]}_{\tau =0}. \end{gathered}$$

Again, using the analogues to (31) for the C derivative, then differentiating $m$ times in the classical sense, and deleting each time a factor, we get

$${{{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }\mathrm{\Gamma }\left(m-\nu \right)}{\mathrm{\Gamma }\left(m-\nu \right)}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(p\right)}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\left[{\displaystyle \frac{{\left(t-\tau \right)}^{m-\nu -1-r\alpha -m}}{\Gamma \left(m-\nu -r\alpha -m\right)}}\right]}_{\tau =0}$$

After some further modifications, we obtain the final result

$${{{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)={\displaystyle \frac{2\mathrm{\pi }}{t^{\nu +1}}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(\frac{p}{t}\right)}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)\Gamma \left(-\nu -r\alpha \right)}},$$

which is the same as (32) and (33). This unification is achieved due to the role of the delta function. □

Corollary 1. 

Show that ${{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }\left(\mathrm{t}\right)\mathrm{\zeta }\left(\mathrm{t}\right)={{}_0{}^{R}D}_t^{\nu }\mathrm{\Gamma }\left(\mathrm{t}\right)\mathrm{\zeta }\left(\mathrm{t}\right)$.

Proof. 

Taking $\alpha =1,$ in (28), we get

$${{{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }}_1\left(\mathrm{t}\right){\mathrm{\zeta }}_1\left(\mathrm{t}\right)={{}_0{}^C\mathrm{D}}_{\mathrm{t}}^{\nu }\mathrm{\Gamma }\left(\mathrm{t}\right)\mathrm{\zeta }\left(\mathrm{t}\right)={{}_0{}^{R}D}_t^{\nu }\mathrm{\Gamma }\left(\mathrm{t}\right)\mathrm{\zeta }\left(\mathrm{t}\right)={\displaystyle \frac{2\mathrm{\pi }}{t^{\nu +1}}}\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(\frac{p}{t}\right)}^r}{p!r!\Gamma \left(-\nu -r\right)}},$$

the required result. □

Next, we consider the Laplace transformation (LT) of the Riemann zeta function employing its new series form.

Theorem 3. 

Find the closed form of the Laplace transform of the new representation, such as

$$\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);\mathrm{s}\right)=2\pi \sum _{p=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\mathrm{\Psi }_1\left[\begin{array}{c}\left(1,1\right)\\ \left(1,\mathrm{\alpha }\right)\end{array};({p\mathrm{s})}^{\mathrm{\alpha }}\right],$$

(34)

and deduce the result for the LT of $\mathrm{\Gamma }\left(\mathrm{\omega }\right)\mathrm{\zeta }\left(\mathrm{\omega }\right)$.

Proof. 

We know that (Table 2 in ref. [14])

$$\mathrm{L}\left\{{\mathrm{\delta }}^{\left(\mathrm{\nu }\right)}\left(\mathrm{\omega }\right);\mathrm{s}\right\}={\mathrm{s}}^{\mathrm{\nu }};\nu \ge 0,$$

consequently, we compute

$$\begin{array}{cc}\hfill \mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);\mathrm{s}\right)=\mathrm{L}& \left(2\mathrm{\pi }{\displaystyle \sum _{\mathrm{n},p,r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\delta }^{\left(r\alpha \right)}\left(\mathrm{\omega }\right);\mathrm{s}\right)\hfill \\ \hfill =2\mathrm{\pi }& {\displaystyle \sum _{\mathrm{n},p,r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}\mathrm{L}\left({\mathrm{\delta }}^{\left(\mathrm{\alpha }\mathrm{r}\right)}\left(\mathrm{\omega }\right);\mathrm{s}\right)\hfill \\ \hfill =2\pi {\displaystyle \sum _{\mathrm{n},p,r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}& {\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}}=2\pi {\displaystyle \sum _{\mathrm{n},\mathrm{p}=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{\mathrm{E}}_{\mathrm{\alpha }}\left(({p\mathrm{s})}^{\mathrm{\alpha }}\right). \hfill \end{array}$$

(35)

Using (15) in (35), we achieve (34). Taking $\mathrm{\alpha }=1$ in (35) gives, (Equation (26) in ref. [10])

$$\mathrm{L}\left\{\mathrm{\Gamma }\left(\mathrm{\omega }\right)\mathrm{\zeta }\left(\mathrm{\omega }\right);\mathrm{s}\right\}=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{\mathrm{e}}^{ps}={\displaystyle \frac{2\mathrm{\pi }}{\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathrm{e}}^{\mathrm{s}}\right)-1}}.$$

(36)

As a result, it authenticates that the findings acquired by the fractional representation are unswerving with the known outcomes, and we move forward to the following result. □

Using the result of Theorem 3, we solve the following fractional differential equation.

Theorem 4. 

For given constants  ${\mathrm{{\rm X}}}_0,\mathrm{c},\mathrm{\nu }>0$, compute the fractional generalized (distributional) solution of the fractional differential equation containing a singular function

$$\mathrm{{\rm X}}\left(\mathrm{t}\right)-{\mathrm{{\rm X}}}_0{\mathrm{\Gamma }}_{\alpha }\left(\mathrm{t}\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{t}\right)=-c^{\nu }{\mathrm{D}}_0^{-\nu }\mathrm{{\rm X}}\left(\mathrm{t}\right);0<\mathrm{\alpha }\le 1,$$

(37)

where ${\mathrm{D}}_0^{-\nu }\mathrm{{\rm X}}\left(\mathrm{t}\right)$ is the R–L fractional derivative defined by (Equation (1) in ref. [30])

$${\mathrm{D}}_0^{-\nu }\mathrm{{\rm X}}\left(\mathrm{t}\right)={{}_0{}^R\mathrm{I}}_{\mathrm{t}}^{\nu }\mathrm{{\rm X}}\left(\mathrm{t}\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{\nu -1}\mathrm{{\rm X}}\left(\mathrm{t}\right)d\tau .$$

Proof. 

For $\mathfrak{R}\left(\mathrm{\omega }\right)>0,$ we use the LT i-e $\mathrm{L}\left[\mathrm{{\rm X}}\left(\mathrm{t}\right);\mathrm{\omega }\right]={\int }_{0^{-}}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{\omega }\mathrm{t}}\mathrm{{\rm X}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}=\mathrm{{\rm X}}\left(\mathrm{\omega }\right)$ over (37)

$$\mathrm{L}\left\{\mathrm{{\rm X}}\left(\mathrm{t}\right);\mathrm{\omega }\right\}-{\mathrm{{\rm X}}}_0\mathrm{L}\left\{{\mathrm{\Gamma }}_{\alpha }\left(\mathrm{t}\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{t}\right);\mathrm{\omega }\right\}=\mathrm{L}\left\{-c^{\mathrm{\nu }}{\mathrm{D}}_0^{-\nu }\mathrm{{\rm X}}\left(\mathrm{t}\right);\mathrm{\omega }\right\},$$

(38)

then, using (35) and (Equation (14) in ref. [14])

$$\mathrm{L}\left\{{\mathrm{D}}_0^{-\nu }\mathrm{{\rm X}}\left(\mathrm{t}\right);\mathrm{\omega }\right\}={\mathrm{\omega }}^{-\mathrm{\nu }}\mathrm{{\rm X}}\left(\mathrm{\omega }\right),$$

(39)

we obtain

$$\mathrm{{\rm X}}\left(\mathrm{\omega }\right)=2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{\omega }}^{\mathrm{\alpha }\mathrm{q}}-{\left({\displaystyle \frac{\mathrm{\omega }}{c}}\right)}^{-\mathrm{\nu }}\mathrm{{\rm X}}\left(\mathrm{\omega }\right).$$

(40)

It should be noted that the Riemann zeta function’s LT is conceivable, only because of its new fractional form (22), which entails

$$\mathrm{{\rm X}}\left(\mathrm{\omega }\right)\left[1+{\left({\displaystyle \frac{\mathrm{\omega }}{c}}\right)}^{-\mathrm{\nu }}\right]=2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{\omega }}^{\mathrm{\alpha }\mathrm{r}}.$$

(41)

After some simple calculations, one can obtain

$$\mathrm{{\rm X}}\left(\omega \right)=2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{\omega }}^{\mathrm{\alpha }\mathrm{r}}\sum _{\mathrm{m}=0}^{\mathrm{\infty }}{\left[-{\left({\displaystyle \frac{\mathrm{\omega }}{c}}\right)}^{-\mathrm{\nu }}\right]}^{\mathrm{m}}.$$

(42)

Taking inverse LT of (42), the required solution is obtained considering the following three cases.

• Case 1: Classical solution valid over the half real line $\mathbf{\nu }\mathbf{m}-\mathbf{\alpha }\mathbf{r}>0$.

Taking inverse LT of (42) using formula (Equation (1) in ref. [14])

$${\mathrm{L}}^{-1}\left\{{\mathrm{s}}^{-(\mathrm{\nu }\mathrm{m}-\mathrm{\alpha }\mathrm{r})};\mathrm{t}\right\}={\displaystyle \frac{{\mathrm{t}}^{\mathrm{\nu }\mathrm{m}-\mathrm{\alpha }\mathrm{r}-1}}{\mathrm{\Gamma }\left(\mathrm{\nu }\mathrm{m}-\mathrm{\alpha }\mathrm{r}\right)}};\mathrm{\nu }\mathrm{m}-\mathrm{\alpha }\mathrm{r}\in {\mathbb{R}}^{+},$$

will imply the required solution using (8) as follows:

$$\begin{array}{cc}\hfill \mathrm{{\rm X}}\left(\mathrm{t}\right)& =2\mathrm{\pi }{\mathrm{{\rm X}}}_0{\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma \left(r\alpha +1\right)}}{\mathrm{t}}^{-\mathrm{\alpha }\mathrm{r}-1}\times {\displaystyle \sum _{\mathrm{m}=0}^{\mathrm{\infty }}}\begin{array}{c}{\displaystyle \frac{{\left(-c^{\mathrm{\nu }}{\mathrm{t}}^{\mathrm{\nu }}\right)}^{\mathrm{m}}}{\mathrm{\Gamma }\left(\mathrm{\nu }\mathrm{m}-\mathrm{\alpha }\mathrm{r}\right)}}\\ .\end{array}\hfill \\ & ={\displaystyle \frac{2\mathrm{\pi }{\mathrm{{\rm X}}}_0}{\mathrm{t}}}{\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-\left(n+1\right)\right)}^p{\left(\frac{p}{t}\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{E}}_{\mathrm{\nu },-\mathrm{\alpha }\mathrm{r}}\left(-c^{\mathrm{\nu }}{\mathrm{t}}^{\mathrm{\nu }}\right); \nu m-\alpha r\ge 1.\hfill \end{array}$$

(43)

The process to compute the solution is conventional, and output function, $\mathrm{{\rm X}}\left(\mathrm{t}\right)$, is naturally described using the Mittag-Leffler function. Infinite summation over the coefficients ${\mathrm{C}}_{\mathrm{\alpha }}\left(\mathrm{t}\right)$ in (43) is well-defined and finite.

$${\mathrm{C}}_{\mathrm{\alpha }}\left(\mathrm{t}\right)=\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(-\frac{p}{t}\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}=\sum _{\mathrm{n},p=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{\mathrm{E}}_{\alpha }\left(-{\displaystyle \frac{\mathrm{p}}{\mathrm{t}}}\right).$$

(44)

Similarly,

$$\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }{\mathrm{C}}_{\mathrm{\alpha }}\left(\mathrm{t}\right)={\displaystyle \frac{1}{e-1}}.$$

(45)

It is worth noting that this solution has fascinating specific situations where α = 1 and vice versa. Hence, we extend the existence of a classical solution over the whole real line using singular entities that are best described using the Dirac delta function. Therefore, we consider the following.

• Case 2: Fractional generalized (distributional) solution

It was proved that (Equation (30) in ref. [14])

$${\mathrm{L}}^{-1}\left\{{\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}};\mathrm{t}\right\}={\mathrm{\delta }}^{\left(\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}\right)}\left(t\right);\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}\ge 0$$

(46)

and using this in (42) implies that

$$\mathrm{{\rm X}}\left(t\right)=2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{m},\mathrm{n},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(p\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{(-\mathrm{c})}^{\mathrm{\nu }\mathrm{m}}{\mathrm{\delta }}^{(\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m})}(t),$$

(47)

which is a generalized (distributional) solution defined over a suitably selected function $f\left(t\right)$ such that

$$\langle \mathrm{{\rm X}}\left(t\right),f\left(t\right)\rangle =2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{m},\mathrm{n},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(p\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{(-\mathrm{c})}^{\mathrm{\nu }\mathrm{m}}\langle {\mathrm{\delta }}^{(\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m})}\left(t\right),f\left(t\right)\rangle .$$

(48)

Then, using Equation (4) in the above equation, we get

$$\langle \mathrm{{\rm X}}\left(\mathrm{t}\right),f\left(t\right)\rangle =2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{m},\mathrm{n},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(p\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\left(-\mathrm{c}\right)}^{\mathrm{\nu }\mathrm{m}}{\left(-1\right)}^{\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}}{f}^{\left(\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}\right)}\left(0\right),$$

(49)

which after some modification yields

$$\langle \mathrm{{\rm X}}\left(\mathrm{t}\right),f\left(t\right)\rangle =2\mathrm{\pi }{\mathrm{{\rm X}}}_0\sum _{\mathrm{m},\mathrm{n},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(-p\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{c}^{\mathrm{\nu }\mathrm{m}}{f}^{(\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m})}(0).$$

(50)

This fractional generalized (distributional) solution (50) is eloquent when $f^{(\mathrm{\nu }\mathrm{m}-\mathrm{\alpha }\mathrm{r})}\left(0\right)$ exists and is convergent. Case 2 cannot be discussed using the existing representation but is possible only due to the new representation explored in this research. □

Remark 2. 

It can be noted that Section 2 is concerned only with the Laplace transform of the new representation. However, more advanced fractional transforms are discussed in the subsequent Section 3.

2.2. Solution of Fractional Differential Models with Singular Kernels

Fractional calculus relies heavily on fractional integrals and derivatives of special functions, which have several applications. We provide easier examples and fractional calculus visualizations of the Riemann zeta function using the following identity (Equation (27) in ref. [29], p. 9)

$$\begin{array}{c}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left\{{\mathrm{s}}^r\right\}={\displaystyle \prod _{\mathrm{j}=1}^{\mathrm{m}}}{\displaystyle \frac{\mathrm{\Gamma }\left({\mathrm{\gamma }}_k+1+\frac{\mathrm{p}}{{\mathrm{\beta }}_k}\right)}{\mathrm{\Gamma }\left({\mathrm{\gamma }}_k+{\nu }_k+1+\frac{\mathrm{p}}{{\mathrm{\beta }}_k}\right)}}{s}^r;\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}({\nu }_{\mathrm{k}}\ge 0;\mathrm{k}=1,\dots ,\mathrm{m};\left[-{\mathrm{\beta }}_{\mathrm{k}}\left({1+\mathrm{\gamma }}_{\mathrm{k}}\right)\right]<\mathrm{r}).\end{array}$$

(51)

This section is further subdivided into two subsections and is concerned with new fractional formulations involving the Riemann zeta function.

2.2.1. An Application of m-E–K Fractional Integral Transforms

Here, new fractional integral formulae using m-E–K integral transforms are computed.

Theorem 5. 

m-E–K integral transform, including the Riemann zeta function, is formulated as

$$\begin{array}{c}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right) ,\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);\mathrm{s}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_{\mathrm{m}}\mathrm{\Psi }_{\mathrm{m}+1}\left[\begin{array}{cc}-& {\left({\mathrm{\gamma }}_k+1+{\displaystyle \frac{\mathrm{\chi }-1}{{\mathrm{\beta }}_k}},{\displaystyle \frac{\mathrm{\alpha }}{{\mathrm{\beta }}_k}}\right)}_1^{\mathrm{m}}\\ (1,\mathrm{\alpha })& {\left({\mathrm{\gamma }}_k+{\nu }_k+1+{\displaystyle \frac{\mathrm{\chi }-1}{{\mathrm{\beta }}_k}},{\displaystyle \frac{\mathrm{\alpha }}{{\mathrm{\beta }}_k}}\right)}_1^{\mathrm{m}}\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]\\ (\mathrm{k}=1,\dots ,\mathrm{m};{\nu }_{\mathrm{k}}\ge 0;\mathrm{p}>\left[-{\mathrm{\beta }}_{\mathrm{k}}\left({1+\mathrm{\gamma }}_{\mathrm{k}}\right)\right])\end{array}$$

(52)

Proof. 

Let us first consider the following using (22)

$${\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);\mathrm{s}\right)\right)={\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left({\mathrm{s}}^{\mathrm{\chi }-1}2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{\omega }}^{\mathrm{\alpha }\mathrm{r}}\right),$$

(53)

then exchanging the integration and summation

$${\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);\mathrm{s}\right)\right)=2\mathrm{\pi }\sum _{\mathrm{p},\mathrm{r}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p({\mathrm{p})}^{\mathrm{\alpha }\mathrm{r}}}{\mathrm{p}!\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{r}+1\right)}}{\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left({{\mathrm{s}}^{\mathrm{\chi }-1}\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}}\right),$$

(54)

and then using (51), it leads to

$${\mathrm{I}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{\left({\mathrm{\gamma }}_{\mathrm{k}}\right),\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);\mathrm{s}\right)\right)=2\mathrm{\pi }\sum _{\mathrm{p},\mathrm{r}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p({\mathrm{p})}^{\mathrm{\alpha }\mathrm{r}}}{\mathrm{p}!\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{r}+1\right)}}\prod _{\mathrm{k}=1}^{\mathrm{m}}{\displaystyle \frac{\mathrm{\Gamma }\left({\mathrm{\gamma }}_k+1+\frac{\mathrm{\chi }+\mathrm{\alpha }\mathrm{r}-1}{{\mathrm{\beta }}_k}\right)}{\mathrm{\Gamma }\left({\mathrm{\gamma }}_k+{\nu }_k+1+\frac{\mathrm{\chi }+\mathrm{\alpha }\mathrm{r}-1}{{\mathrm{\beta }}_k}\right)}}{\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}+\mathrm{\chi }-1},$$

(55)

which, after an application of (13), becomes the required simplified form in terms of the Fox–Wright function. □

Table 2. New fractional integrals using delta function.

m = 3	M–S–M Fractional Integrals
${\mathrm{I}}_{0+}^{{\mathrm{\gamma }}_1,{{\mathrm{\gamma }}_1}^{\prime },{\mathrm{\gamma }}_2,{{\mathrm{\gamma }}_2}^{\prime },\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\nu }+\mathrm{\chi }-{\mathrm{\gamma }}_1-{{\mathrm{\gamma }}_1}^{\prime }-1}$ ${\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_3\mathrm{\Psi }_4\left[\begin{array}{cccc}-& \left(\mathrm{\chi },\mathrm{\alpha }\right)& \left(\mathrm{\chi }+\mathrm{\nu }-{\mathrm{\gamma }}_1-{{\mathrm{\gamma }}_1}^{\prime }-{\mathrm{\gamma }}_2,\mathrm{\alpha }\right)& \left(\mathrm{\chi }+{{\mathrm{\gamma }}_2}^{\prime }-{{\mathrm{\gamma }}_1}^{\prime },\mathrm{\alpha }\right)\\ (1,\mathrm{\alpha })& \left(\mathrm{\chi }+{{\mathrm{\gamma }}_2}^{\prime },\mathrm{\alpha }\right)& \left(\mathrm{\chi }+\mathrm{\nu }-{\mathrm{\gamma }}_1-{{\mathrm{\gamma }}_1}^{\prime },\mathrm{\alpha }\right)& (\mathrm{\chi }+\mathrm{\nu }-{{\mathrm{\gamma }}_1}^{\prime }-{\mathrm{\gamma }}_2,\mathrm{\alpha })\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]$
${\mathrm{I}}_{0-}^{{\mathrm{\gamma }}_1,{{\mathrm{\gamma }}_1}^{\prime },{\mathrm{\gamma }}_2,{{\mathrm{\gamma }}_2}^{\prime },\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\nu }+\mathrm{\chi }-{\mathrm{\gamma }}_1-{{\mathrm{\gamma }}_1}^{\prime }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_3\mathrm{\Psi }_4$ $\left[\begin{array}{cccc}-& \left(1-\mathrm{\chi }-\mathrm{\nu }+{\mathrm{\gamma }}_1+{{\mathrm{\gamma }}_1}^{\prime },\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }+{\mathrm{\gamma }}_1+{{\mathrm{\gamma }}_2}^{\prime }-\mathrm{\nu },\mathrm{\alpha }\right)& (1-\mathrm{\chi }-{\mathrm{\gamma }}_1,\mathrm{\alpha })\\ (1,\mathrm{\alpha })& \left(1-\mathrm{\chi },\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }+{\mathrm{\gamma }}_1+{{\mathrm{\gamma }}_1}^{\prime }+{{\mathrm{\gamma }}_2}^{\prime }-\mathrm{\nu },\mathrm{\alpha }\right)& (1-\mathrm{\chi }+{\mathrm{\gamma }}_1-{\mathrm{\gamma }}_2,\mathrm{\alpha })\end{array}|{\left({\displaystyle \frac{\mathrm{p}}{\mathrm{s}}}\right)}^{\mathrm{\alpha }}\right]$
m = 2	Saigo fractional integrals
	$I_{0+}^{{\gamma }_1,{\gamma }_2,\nu }\left({\omega }^{\chi -1}L\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\pi s^{\chi -{\gamma }_1-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_2\Psi _3\left[\begin{array}{ccc}-& \left(\chi ,\alpha \right)& \left(\chi +{\gamma }_2-{\gamma }_1,\alpha \right)\\ (1,\alpha )& \left(\chi -{\gamma }_2,\alpha \right)& \left(\chi +\nu +{\gamma }_2,\alpha \right)\end{array}|{\left(ps\right)}^{\alpha }\right]$
	$I_{-}^{{\gamma }_1,{\gamma }_2,\nu }\left({\omega }^{\chi -1}L{(\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)=2\pi s^{\chi -{\gamma }_1-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_2\Psi _3\left[\begin{array}{ccc}-& \left({\gamma }_1-\chi +1,\alpha \right)& \left({\gamma }_2-\chi +1,\alpha \right)\\ (1,\alpha )& \left(1-\chi ,\alpha \right)& ({\gamma }_1+{\gamma }_2+\nu -\chi +1,\alpha )\end{array}|{\left({\displaystyle \frac{p}{s}}\right)}^{\alpha }\right]$
m = 1	E–K fractional integrals
	${}_{0+}I_{\beta }^{\gamma ,\nu } \left({\omega }^{\chi -1}L\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\pi s^{\chi -1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\Psi _2\left[\begin{array}{cc}-& \mathrm{\gamma }+1+{\displaystyle \frac{\mathrm{\chi }-1}{\mathrm{\beta }}},{\displaystyle \frac{\mathrm{\alpha }}{\mathrm{\beta }}}\\ (1,\mathrm{\alpha })& \mathrm{\gamma }+\mathrm{\nu }+1+{\displaystyle \frac{\mathrm{\chi }-1}{\mathrm{\beta }}},{\displaystyle \frac{\mathrm{\alpha }}{\mathrm{\beta }}}\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]$
	${}_{-}I_{\beta }^{\gamma ,\nu } \left({\omega }^{\chi -1}L\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right)\right);{\displaystyle \frac{1}{s}}\right)=2\pi s^{\chi -1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\Psi _2\left[\begin{array}{cc}-& \mathrm{\gamma }-{\displaystyle \frac{\mathrm{\chi }-1}{\mathrm{\beta }}},{\displaystyle \frac{\mathrm{\alpha }}{\mathrm{\beta }}}\\ (1,\mathrm{\alpha })& \mathrm{\gamma }+\mathrm{\nu }-{\displaystyle \frac{\mathrm{\chi }-1}{\mathrm{\beta }}},{\displaystyle \frac{\mathrm{\alpha }}{\mathrm{\beta }}}\end{array}|{\left({\displaystyle \frac{p}{s}}\right)}^{\alpha }\right]$
m = 1	R- L fractional integrals
	${\mathrm{I}}_{0+}^{\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }+\mathrm{\nu }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\mathrm{\Psi }_2\left[\begin{array}{cc}-& \left(\mathrm{\chi },\mathrm{\alpha }\right)\\ (1,\mathrm{\alpha })& \left(\mathrm{\nu }+\mathrm{\chi },\mathrm{\alpha }\right)\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]$
	${\mathrm{I}}_{-}^{\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }+\mathrm{\nu }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\mathrm{\Psi }_2\left[\begin{array}{cc}-& \left(1-\mathrm{\nu }-\mathrm{\chi },\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(1-\mathrm{\chi },\mathrm{\alpha }\right)\end{array}|{\left({\displaystyle \frac{\mathrm{p}}{\mathrm{s}}}\right)}^{\mathrm{\alpha }}\right]$

includes other relevant exceptional instances. Although comparable methods and outcomes are used for the RHS integrals, we focused on the results for LHS integrals in the preceding sections. Table 2 also takes into account some popular fractional integrals of the R–L type. The fractional integrals on the LHS and RHS are represented using the notations $I_{0+}$, $I_{0-}$ respectively in Table 2, as is standard practice.

It is worth noting that fascinating, exceptional situations can be achieved with $\mathrm{\alpha }=1$, and vice versa.

2.2.2. An Application of m-E–K Fractional Derivatives

We can derive its generalized fractional derivatives using Theorem 1′s methodology and a novel depiction of the Riemann zeta function. Here, we obtain them directly by utilizing the general conclusion (Theorem 4 in ref. [29]), which is given as

$${\mathrm{D}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{{\left({\mathrm{\gamma }}_{\mathrm{k}}\right)}_1^{\mathrm{m}},\left({\mathrm{\nu }}_{\mathrm{k}}\right)}\left\{{\mathrm{z}}^{\mathrm{c}}{}_{\mathrm{p}}\mathrm{\Psi }_{\mathrm{q}}\left[\begin{array}{c}{\left({\mathrm{a}}_{\mathrm{i}},{\mathrm{\alpha }}_{\mathrm{i}}\right)}_1^{\mathrm{p}}\\ {{(\mathrm{b}}_{\mathrm{j}},{\mathrm{\beta }}_{\mathrm{j}})}_1^{\mathrm{q}}\end{array};\mathrm{\lambda }{\mathrm{z}}^{\mathrm{\mu }}\right]\right\}={\mathrm{z}}^{\mathrm{c}}\left\{{}_{\mathrm{p}+\mathrm{m}}\mathrm{\Psi }_{\mathrm{q}+\mathrm{m}}\left[\begin{array}{c}{\left({\mathrm{a}}_{\mathrm{i}},{\mathrm{\alpha }}_{\mathrm{i}}\right)}_1^{\mathrm{p}}{,\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\nu }}_{\mathrm{k}}+1+{\displaystyle \frac{\mathrm{c}}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\\ {{(\mathrm{b}}_{\mathrm{j}},{\mathrm{\beta }}_{\mathrm{j}})}_1^{\mathrm{q}}{,\left({\mathrm{\gamma }}_{\mathrm{k}}+1+{\displaystyle \frac{\mathrm{c}}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{1}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\end{array};\mathrm{\lambda }{\mathrm{z}}^{\mathrm{\mu }}\right]\right\}$$

(56)

It was mentioned that ([29], p. 2) “we are not capable of studying Euler gamma and Riemann zeta functions, even if we intend to include the largest family of special functions”. In view of this remark by Kiryakova [29], it is significant to mention that fractional derivatives containing these functions are attained by using m-E–K fractional derivatives in this research, as follows:

$${\mathrm{D}}_{\left({\mathrm{\beta }}_{\mathrm{k}}\right),\mathrm{m}}^{{\left({\mathrm{\gamma }}_{\mathrm{k}}\right)}_1^{\mathrm{m}},\left({\mathrm{\nu }}_k\right)}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right)\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }-1}\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_{\mathrm{m}}\mathrm{\Psi }_{\mathrm{m}+1}\left[\begin{array}{cc}-& {\left({\mathrm{\gamma }}_{\mathrm{k}}+{\mathrm{\nu }}_{\mathrm{k}}+1+{\displaystyle \frac{\mathrm{\chi }}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{\mathrm{\alpha }}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\\ \left(1,\mathrm{\alpha }\right)& {\left({\mathrm{\gamma }}_{\mathrm{k}}+1+{\displaystyle \frac{\mathrm{\chi }}{{\mathrm{\beta }}_{\mathrm{k}}}},{\displaystyle \frac{\mathrm{\alpha }}{{\mathrm{\beta }}_{\mathrm{k}}}}\right)}_1^{\mathrm{m}}\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right].$$

(57)

This is a novel achievement attained by the new representation. Further important special cases are presented in the subsequent

Table 3. New fractional derivatives using delta function.

m = 3	M–S–M Fractional Derivatives
${\mathrm{D}}_{0+}^{{\mathrm{\gamma }}_1,{{\mathrm{\gamma }}_1}^{\prime },{\mathrm{\gamma }}_2,{{\mathrm{\gamma }}_2}^{\prime },\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }+{\mathrm{\gamma }}_1+{{\mathrm{\gamma }}_1}^{\prime }-\mathrm{\nu }-1}$ ${\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_3\mathrm{\Psi }_4\left[\begin{array}{cccc}-& \left(\mathrm{\chi },\mathrm{\alpha }\right)& \left(\mathrm{\chi }-{\mathrm{\gamma }}_2+{\mathrm{\gamma }}_1,\mathrm{\alpha }\right)& \left(\mathrm{\chi }+{\mathrm{\gamma }}_{1+}{{\mathrm{\gamma }}_1}^{\prime }{{+\mathrm{\gamma }}_2}^{\prime }-\mathrm{\nu },\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(\mathrm{\chi }-{\mathrm{\gamma }}_2,\mathrm{\alpha }\right)& \left(\mathrm{\chi }-\mathrm{\nu }+{\mathrm{\gamma }}_1+{{+\mathrm{\gamma }}_2}^{\prime },\mathrm{\alpha }\right)& \left(\mathrm{\chi }-\mathrm{\nu }+{{\mathrm{\gamma }}_1}^{\prime }+{\mathrm{\gamma }}_1,\mathrm{\alpha }\right)\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]$
${\mathrm{D}}_{-}^{{\mathrm{\gamma }}_1,{{\mathrm{\gamma }}_1}^{\prime },{\mathrm{\gamma }}_2,{{\mathrm{\gamma }}_2}^{\prime },\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }+{\mathrm{\gamma }}_1+{{\mathrm{\gamma }}_1}^{\prime }-\mathrm{\nu }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}$ ${}_3\mathrm{\Psi }_4\left[\begin{array}{cccc}-& \left(1-\mathrm{\chi }{{+\mathrm{\gamma }}_2}^{\prime },\mathrm{\alpha }\right)& \left(1{{+\mathrm{\gamma }}_1}^{\prime }-\mathrm{\chi }-{\mathrm{\gamma }}_2+\mathrm{\nu },\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }-{\mathrm{\gamma }}_{1-}{{\mathrm{\gamma }}_1}^{\prime }+\mathrm{\nu },\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(1-\mathrm{\chi },\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }{{-\mathrm{\gamma }}_1}^{\prime }{{+\mathrm{\gamma }}_2}^{\prime },\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }+\mathrm{\nu }-{{\mathrm{\gamma }}_1}^{\prime }-{\mathrm{\gamma }}_1-{\mathrm{\gamma }}_2,\mathrm{\alpha }\right)\end{array}|{\left({\displaystyle \frac{\mathrm{p}}{\mathrm{s}}}\right)}^{\mathrm{\alpha }}\right]$
m = 2	Saigo fractional derivatives
	${\mathrm{D}}_{0+}^{{\mathrm{\gamma }}_1,{\mathrm{\gamma }}_2,\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }+{\mathrm{\gamma }}_1-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}$ ${}_2\mathrm{\Psi }_3\left[\begin{array}{ccc}-& \left(\mathrm{\chi },\mathrm{\alpha }\right)& \left(\mathrm{\chi }+\mathrm{\nu }+{\mathrm{\gamma }}_2+{\mathrm{\gamma }}_1,\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(\mathrm{\chi }+{\mathrm{\gamma }}_2,\mathrm{\alpha }\right)& \left(\mathrm{\chi }+\mathrm{\nu },\mathrm{\alpha }\right)\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]$
	${\mathrm{D}}_{-}^{{\mathrm{\gamma }}_1,{\mathrm{\gamma }}_2,\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }+{\mathrm{\gamma }}_1-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}$ ${}_2\mathrm{\Psi }_3\left[\begin{array}{ccc}-& \left(1-\mathrm{\chi }-{\mathrm{\gamma }}_2,\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }+\mathrm{\nu }+{\mathrm{\gamma }}_1,\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(1-\mathrm{\chi }+\mathrm{\nu }-{\mathrm{\gamma }}_2,\mathrm{\alpha }\right)& \left(1-\mathrm{\chi },\mathrm{\alpha }\right)\end{array}|{\left({\displaystyle \frac{\mathrm{p}}{\mathrm{s}}}\right)}^{\mathrm{\alpha }}\right]$
m = 1	E–K fractional derivatives
	${}_{0+}D_{\beta }^{\gamma ,\nu }\left({\omega }^{\chi -1}L\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\pi s^{\chi -1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\Psi _2\left[\begin{array}{cc}-& \gamma +\nu +1+{\displaystyle \frac{\chi -1}{\beta }},{\displaystyle \frac{\alpha }{\beta }}\\ (1,\alpha )& \gamma +1+{\displaystyle \frac{\chi -1}{\beta }},{\displaystyle \frac{\alpha }{\beta }}\end{array}|{\left(ps\right)}^{\alpha }\right]$
	${}_{-}D_{\beta }^{\gamma ,\nu }\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\Psi _2\left[\begin{array}{cc}-& \mathrm{\gamma }+\mathrm{\nu }-{\displaystyle \frac{\mathrm{\chi }-1}{\mathrm{\beta }}},{\displaystyle \frac{\mathrm{\alpha }}{\mathrm{\beta }}}\\ (1,\mathrm{\alpha })& \mathrm{\gamma }-{\displaystyle \frac{\mathrm{\chi }-1}{\mathrm{\beta }}},{\displaystyle \frac{\mathrm{\alpha }}{\mathrm{\beta }}}\end{array}|{\left({\displaystyle \frac{\mathrm{p}}{\mathrm{s}}}\right)}^{\mathrm{\alpha }}\right]$
m = 1	R–L fractional derivatives
	${\mathrm{D}}_{0+}^{\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);s\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }-\mathrm{\nu }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\mathrm{\Psi }_2\left[\begin{array}{cc}-& \left(\mathrm{\chi },\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(\mathrm{\chi }-\mathrm{\nu },\mathrm{\alpha }\right)\end{array}|{\left(\mathrm{p}\mathrm{s}\right)}^{\mathrm{\alpha }}\right]$
	${\mathrm{D}}_{-}^{\mathrm{\nu }}\left({\mathrm{\omega }}^{\mathrm{\chi }-1}\mathrm{L}\left({\mathrm{\Gamma }}_{\alpha }\left(\mathrm{\omega }\right){\mathrm{\zeta }}_{\alpha }\left(\mathrm{\omega }\right);{\displaystyle \frac{1}{s}}\right)\right)=2\mathrm{\pi }{\mathrm{s}}^{\mathrm{\chi }-\mathrm{\nu }-1}{\displaystyle \sum _{n,p=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\mathrm{\Psi }_2\left[\begin{array}{cc}-& \left(\mathrm{\nu }-\mathrm{\chi }+1,\mathrm{\alpha }\right)\\ \left(1,\mathrm{\alpha }\right)& \left(1-\mathrm{\chi },\mathrm{\alpha }\right)\end{array}|{\left({\displaystyle \frac{\mathrm{p}}{\mathrm{s}}}\right)}^{\mathrm{\alpha }}\right]$

. This also takes into account some popular fractional derivatives of the R–L type. The fractional derivatives on the LHS and RHS are represented using the notations $D_{0+}$, $D_{0-}$ respectively, in Table 3, as this is standard practice.

It is worth noting that fascinating exceptional situations can be achieved with $\mathrm{\alpha }=1$, and vice versa.

3. Discussion and Further Examples

It can be noted that Section 2 is concerned with the standard transforms. However, in this section, we explore further the new representation by considering the miscellaneous examples. This exemplification of the Riemann zeta function is an infinite summation over delta function, which is certain only if defined in the way of distributions (generalized functions). Because the delta function is a linear mapping, it is capable of mapping each function to its zero value. Given this information, we study the subsequent examples. Assume (22) and confine the variable, $u=t$, to real numbers. Then, we obtain

$${\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)=2\mathrm{\pi }\sum _{\mathrm{p},\mathrm{r}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p({\mathrm{p})}^{\mathrm{\alpha }\mathrm{r}}}{\mathrm{p}!\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{r}+1\right)}}{\mathrm{\delta }}^{\left(\mathrm{\alpha }\mathrm{r}\right)}\left(\mathrm{t}\right)$$

and

$$\begin{array}{c}\langle {\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right),\mathrm{f}\left(\mathrm{t}\right)\rangle =2\pi {\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}\langle {\mathrm{\delta }}^{\left(\mathrm{\alpha }\mathrm{r}\right)}\left(\mathrm{t}\right),\mathrm{f}\left(\mathrm{t}\right)\rangle \\ =2\pi {\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\left(-1\right)}^{\mathrm{\alpha }\mathrm{r}}{\mathrm{f}}^{\left(\mathrm{\alpha }\mathrm{r}\right)}\left(0\right).\end{array}$$

$$\mathrm{s}\mathrm{u}\mathrm{m} \mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}=\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{\mathrm{E}}_{\mathrm{\alpha }}\left(({-\mathrm{p})}^{\mathrm{\alpha }}\right)=\sum _{n,p=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{}_1\mathrm{\Psi }_1\left[\begin{array}{c}\left(1,1\right)\\ \left(1,\mathrm{\alpha }\right)\end{array};({-\mathrm{p})}^{\mathrm{\alpha }}\right]$$

is finite and well-defined. It involves a number of fractional operators, and evaluating fractional derivatives of even essential functions necessitates a detailed comprehension of the selected operator. Garrappa et al. [33] reviewed the most commonly used operators and presented two techniques for generalizing integer order derivatives to fractional order in order to deliver a way for such comprehension of the unique aspects of each fractional derivative and to better emphasize their peculiarities. In ref. [33], a tutorial on evaluating derivatives of several elementary functions and fractional integrals is provided, and the influence of multiple derivatives on the same function is investigated. Hence, it was notably emphasized that, over long periods of time, the Caputo and R–L derivatives congregate to the Grünwald–Letnikov derivative as a perfect generalization of conventional integer-order derivatives but is rarely advantageous in real-world models [33]. Hence, we consider the following examples by using Grünwald–Letnikov derivative using the same notation as in ref. [33]:

Example 1. 

Consider  $\mathrm{f}\left(\mathrm{t}\right)={\mathrm{e}}^{\mathrm{c}\mathrm{t}}$ so that ${}^{GL}D^{\alpha r}\mathrm{f}\left(\mathrm{t}\right)={\mathrm{c}}^{\mathrm{\alpha }\mathrm{r}}{\mathrm{e}}^{\mathrm{c}\mathrm{t}}.$

Then, using (4) and (22) for this function, we get

$$\langle {\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right),{\mathrm{e}}^{\mathrm{c}\mathrm{t}}\rangle =2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\left(-\mathrm{c}\right)}^{\mathrm{\alpha }\mathrm{r}}=2\mathrm{\pi }\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{\mathrm{p}!}}{\mathrm{E}}_{\mathrm{\alpha }}{\left(-\mathrm{c}\mathrm{p}\right)}^{\mathrm{\alpha }}.$$

Example 2. 

Consider $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{t}$ so that ${}^{GL}D^{\alpha r} f\left(t\right)={\mathrm{c}}^{\mathrm{\alpha }\mathrm{r}}\sin \left(ct+{\displaystyle \frac{\mathrm{\alpha }\mathrm{r}\mathrm{\pi }}{2}}\right)$.

Then, using (4) and (22) for this function, we get

$$\begin{array}{cc}\hfill \langle {\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right),\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{t}\rangle & =2\mathrm{\pi }{\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{\left(cp\right)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\left(-1\right)}^{\mathrm{\alpha }\mathrm{r}}\sin \left({\displaystyle \frac{\mathrm{\alpha }\mathrm{r}\mathrm{\pi }}{2}}\right)=\mathfrak{I}\left(2\mathrm{\pi }{\displaystyle \sum _{\mathrm{n},\mathrm{p},r=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p{(-cp)}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{e}^{i{\displaystyle \frac{\mathrm{\alpha }\mathrm{r}\mathrm{\pi }}{2}}}\right)\hfill \\ & =\mathfrak{I}\left(2\mathrm{\pi }{\displaystyle \sum _{\mathrm{p},\mathrm{r}=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p(-e^{\frac{\mathrm{i}\mathrm{\pi }}{2}}{\mathrm{c}\mathrm{p})}^{\mathrm{\alpha }\mathrm{r}}}{\mathrm{p}!\mathrm{\Gamma }\left(\mathrm{\alpha }\mathrm{r}+1\right)}}\right)=\mathfrak{I}\left(2\mathrm{\pi }{\displaystyle \sum _{\mathrm{p}=0}^{\mathrm{\infty }}}{\displaystyle \frac{{\left(-(n+1\right))}^p}{p!}}{\mathrm{E}}_{\mathrm{\alpha }}{\left(-\mathrm{c}\mathrm{p}{e}^{{\displaystyle \frac{\mathrm{i}\mathrm{\pi }}{2}}}\right)}^{\mathrm{\alpha }}\right).\hfill \end{array}$$

Similarly, $\mathrm{f}\left(\mathrm{t}\right)=\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{t}$ then

$${}^{GL}D^{\alpha r}f\left(t\right)={\mathrm{c}}^{\mathrm{\alpha }\mathrm{r}}\cos \left(ct+{\displaystyle \frac{\mathrm{\alpha }\mathrm{r}\mathrm{\pi }}{2}}\right)$$

$$\langle {\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right),\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{t}\rangle =\mathfrak{R}\left(2\mathrm{\pi }\sum _{\mathrm{p}=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p}{\mathrm{p}!}}{\mathrm{E}}_{\mathrm{\alpha }}{\left(-\mathrm{c}\mathrm{p}{e}^{{\displaystyle \frac{\mathrm{i}\mathrm{\pi }}{2}}}\right)}^{\mathrm{\alpha }}\right).$$

Example 3. 

Formulation of the fractional Volterra integral equation of the first and second kind.

Since the fractional differential and integral equations with singular kernels arise naturally in the mathematical modelling of engineering systems characterized by memory effects, impulsive excitation, and nonlocal temporal behaviour, in particular, distributional formulations involving fractional derivatives of the Dirac delta function have been shown to admit physical interpretations as memory kernels in fractional viscoelastic models, including the Scott–Blair (spring-pot) rheological element [14]. Within this context, the fractional distributional representation developed in the present study provides a rigorous analytical framework for examining fractional models with singular coefficients and long-range memory. Such formulations are relevant to theoretical investigations of viscoelasticity, signal processing, and anomalous transport phenomena [14] due to Dirac delta involvement and applications. Volterra integral equations are important because they allow us to model systems whose future behaviour is influenced by their previous history, which is a common feature in real-world circumstances. These equations are employed in many domains, including population dynamics, epidemic spread, semiconductor devices, and even the analysis of viscoelastic materials. Volterra integral equation of the fractional order is stated as [34]

$${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{\nu -1}\mathrm{y}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }=f\left(\mathrm{t}\right).$$

While the above equation is of the first kind, the second kind is given as

$$\mathrm{y}\left(\mathrm{t}\right)=f\left(\mathrm{t}\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{\nu -1}\mathrm{y}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }.$$

Let us formulate a novel fractional Volterra integral equation containing (22)

$$\mathrm{y}\left(\mathrm{t}\right)={\mathrm{\Gamma }}_{\alpha }\left(t\right){\mathrm{\zeta }}_{\alpha }\left(t\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{\nu -1}\mathrm{y}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau }.$$

Taking the Laplace transform on both sides and using (22), we get

$$\mathrm{Y}\left(\mathrm{s}\right)=2\mathrm{\pi }\sum _{\mathrm{n},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}}+{\displaystyle \frac{\mathrm{Y}\left(\mathrm{s}\right)}{s^{\nu }}}$$

$$\mathrm{Y}\left(\mathrm{s}\right)=2\mathrm{\pi }\sum _{\mathrm{n},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}}{\left[1-{\displaystyle \frac{1}{s^{\nu }}}\right]}^{-1}$$

$$\mathrm{Y}\left(\mathrm{s}\right)=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{m},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}}; \mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}\ge 0.$$

Using the inverse Laplace transform on both sides, we obtain

$$\mathrm{y}\left(\mathrm{t}\right)=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{m},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\mathrm{L}}^{-1}\left[{\mathrm{s}}^{\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}};t\right],$$

and

$$\mathrm{y}\left(\mathrm{t}\right)=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{m},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p({\frac{p}{t})}^{\alpha r}{t}^{\mathrm{\nu }\mathrm{m}}}{p!\Gamma (r\alpha +1)\mathrm{\Gamma }\left(-\mathrm{\alpha }\mathrm{r}+\mathrm{\nu }\mathrm{m}\right)}}.$$

This also holds on the extended domain ($\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}\in \mathbb{R}$); the corresponding distributional solution valid over the space of functions is

$$\mathrm{y}\left(\mathrm{t}\right)=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{m},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^{r\alpha }}{p!\Gamma (r\alpha +1)}}{\delta }^{\left(\mathrm{\alpha }\mathrm{r}-\mathrm{\nu }\mathrm{m}\right)}\left(t\right).$$

For $\alpha =1$, the result holds for the product of gamma and zeta functions as follows:

$$\mathrm{y}\left(\mathrm{t}\right)=\mathrm{\Gamma }\left(t\right)\mathrm{\zeta }\left(t\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\underset{0^{-}}{\overset{t}{\int }}{\left(t-\tau \right)}^{\nu -1}\mathrm{y}\left(\mathrm{\tau }\right)\mathrm{d}\mathrm{\tau },$$

and the solution is

$$\mathrm{y}\left(\mathrm{t}\right)=2\mathrm{\pi }\sum _{\mathrm{n},\mathrm{m},p,r=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-(n+1\right))}^p{p}^r}{p!r!}}{\delta }^{\left(\mathrm{r}-\mathrm{\nu }\mathrm{m}\right)}\left(t\right).$$

Certain other forms of this solution can also be discussed. Similarly, other integral and differential equations can be modelled and solved using these new facts explored in this research.

4. Conclusions and Future Directions

This study introduced a distributional framework for fractional integral and differential equations with singular kernels by constructing a fractional Taylor representation of the product of Euler gamma and Riemann zeta functions based on fractional derivatives of the Dirac delta distribution. The proposed formulation overcomes analytical limitations caused by nonremovable singularities and enables the derivation of new fractional identities through Laplace transformation. Using this representation, several novel fractional integral and derivative formulas involving the gamma and zeta functions were obtained via multiple Erdélyi–Kober operators. A fractional differential equation containing the product of Euler gamma and Riemann zeta functions was formulated and solved analytically. Both classical and generalized distributional solutions were derived, allowing the extension of solutions from the positive real axis to the entire real line—an outcome that is unattainable using conventional integer-order methods.

From a physical application perspective, the developed framework provides a consistent analytical tool for modelling systems with singular memory kernels, impulsive excitation, and long-range temporal effects. Such characteristics commonly arise in viscoelastic materials, anomalous diffusion processes, and signal processing applications involving impulsive responses. The ability to obtain closed-form solutions for these models enhances both theoretical understanding and practical analysis. The results presented in this work are intended primarily as analytical and theoretical contributions. Future work may focus on numerical implementations of the proposed operators, stability analysis of the resulting fractional models, and further applications in quantum mechanics [35], control systems, and transport phenomena where singular fractional kernels play a crucial role. Without asserting direct experimental validation or specific engineering implementation, the proposed framework may be regarded as a mathematically consistent tool for the theoretical analysis of fractional models involving singular memory kernels using recent work [36,37] with the Rathie I-function as a kernel. The closed-form expressions obtained here may assist future studies concerned with the modelling, approximation, or numerical treatment of engineering systems exhibiting nonlocal and hereditary effects.