*Article* **Fractional Integrals Associated with the One-Dimensional Dunkl Operator in Generalized Lizorkin Space**

**Fethi Bouzeffour**

Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia<EMAIL_ADDRESS>
**Abstract:** This paper explores the realm of fractional integral calculus in connection with the onedimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context.

**Keywords:** Dunkl theory; fractional Integral; Bessel functions

**MSC:** 42B30; 33C52; 33C67; 33D67; 33D80; 35K08; 42B25; 42C05

#### **1. Introduction**

On the real line, for a positive real number *κ*, the Dunkl operator D*κ* provides a one-parameter deformation of the ordinary derivative *d dx* . It is defined as:

$$\mathcal{O}\_{\mathbf{x}} := \frac{d}{d\mathbf{x}} + \frac{\kappa}{\mathbf{x}}(1-\mathbf{s}) \,. \tag{1}$$

where *s* is the reflection operator acting on a function *f*(*x*) of a real variable *x* as *s f*(*x*) := *f*(−*x*). The Dunkl operator incorporates the additional term *κ x* (1 − *s*), which accounts for reflection symmetry and introduces a dependence on the parameter *κ*. This operator plays a fundamental role in generalizing various classical results in harmonic analysis and approximation theory, as explored in the works of Dunkl [1,2] Trimeche [3], de Jeu [4], Rosler [5–7], and others.

Fractional calculus [8–15] has gained significant importance in recent decades as a powerful tool for developing advanced mathematical models involving fractional differential and integral operators. When applied to the Dunkl operator, fractional calculus offers a fresh perspective by incorporating the effects of reflection and asymmetry within the underlying space.

A notable feature of the Dunkl setting is the existence of a natural Riesz transform, which shares similarities with classical singular integrals. In the multidimensional case, S. Thangavelu and Y. Xu [16,17] established the *Lp*-boundedness of the associated Riesz transform. This study was further extended by Amri and Sifi [18], who considered the general case for 1 < *p* < ∞. Additionally, investigations into singular integrals and multipliers were carried out in [18–22]. These contributions have significantly enriched our understanding of the Dunkl operator and its associated Riesz transform.

**Citation:** Bouzeffour, F. Fractional Integrals Associated with the One-Dimensional Dunkl Operator in Generalized Lizorkin Space. *Symmetry* **2023**, *15*, 1725. https://doi.org/10.3390/ sym15091725

Academic Editors: Francisco Martínez González and Mohammed K. A. Kaabar

Received: 30 August 2023 Revised: 5 September 2023 Accepted: 6 September 2023 Published: 8 September 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In this study, our main focus is on the comprehensive exploration of the one-dimensional fractional Dunkl integral within Lizorkin type spaces [10–12], with a specific emphasis on analytic continuation techniques. The obtained operators go beyond the conventional Riesz fractional integral [9] and Feller fractional integral [8,11], as they are specifically tailored to operate within the Dunkl setting. By extending the applicability of these operators to the Dunkl context, we aim to unlock new possibilities and gain deeper insights into the realm of fractional calculus.

To address the challenges posed by the divergence of fractional Dunkl operators, we adopt a unique approach that incorporates the regularization technique for divergent integrals, inspired by the work described in the book by Samko [11,12]. Our methodology involves utilizing specific segments of the Taylor formula associated with the Dunkl operator, as originally formulated by Mourou [23]. This regularization technique plays a pivotal role in extending the fractional integral operators to the domain of (*α*) > 0. As a result, we introduce an alternative normalization scheme for tempered power functions, offering a fresh and insightful perspective on fractional calculus within the Dunkl setting. It is important to note that while Soltani [24] relies on the conventional Taylor series, our approach, based on the Taylor formula of Mourou [23], better suits the specific requirements of the Dunkl operator.

Our paper is organized as follows: In Section 2, we begin by collecting some essential facts about the Dunkl operator and the Lizorkin space. Section 3 focuses on studying the generalized power function and its analytic continuation. Moving on to Section 4, we dedicate that section to the study of extensions of well-known fractional integrals such as the Riesz fractional integral, the Feller fractional integral, and the Weyl fractional integral.

#### **2. Preliminaries**

In this section, we introduce some notations and gather some facts about the onedimensional Dunkl operator.

#### *2.1. The One-Dimensional Dunkl Operator*

Let *κ* ≥ 0, and *f* be a differentiable function on R. The Dunkl derivative D*κ f*(*x*) is defined by

$$\mathcal{O}\_{\mathbf{x}}f(\mathbf{x}) = \begin{cases} \ f'(\mathbf{x}) + \kappa \frac{f(\mathbf{x}) - f(-\mathbf{x})}{\mathbf{x}}, & \text{if} \quad \mathbf{x} \neq \mathbf{0}, \\\\ (2\kappa + 1)f'(\mathbf{0}), & \text{if} \quad \mathbf{x} = \mathbf{0}. \end{cases} \tag{2}$$

We denote by *Lp κ* (R) (1 ≤ *p*), the Lebesgue space associated with the measure

$$\sigma\_{\kappa}(d\mathbf{x}) = \frac{|\mathbf{x}|^{2\kappa}}{2^{\kappa+1/2}\Gamma(\kappa+1/2)} \, d\mathbf{x} \tag{3}$$

and by *f κ*,*p* the usual norm given by

$$\|f\|\_{\mathbb{K},p} = \left(\int\_{\mathbb{R}} |f(\mathfrak{x})|^p \, \sigma\_{\mathbb{K}}(d\mathfrak{x})\right)^{1/p}.\tag{4}$$

Now, consider the so-called *nonsymmetric Bessel function*, also called *Dunkl type Bessel function*, in the rank one case (see [25]) [§10.22(v)]:

$$\mathcal{A}\_{\mathbf{x}}(\mathbf{x}) \coloneqq \mathcal{J}\_{\mathbf{x}-1/2}(i\mathbf{x}) + \frac{\mathbf{x}}{2\kappa + 1} \; \mathcal{J}\_{\kappa + 1/2}(i\mathbf{x}). \tag{5}$$

where the normalized Bessel functions is defined by

$$\begin{aligned} \mathcal{J}\_\kappa(\infty) &:= \Gamma(\kappa + 1) \,(2/\chi)^\kappa \, f\_\kappa(\infty) \\ &= \sum\_{n=0}^\infty \frac{(-1)^n}{n! \Gamma(\kappa + n + 1)} (\frac{\chi}{2})^{2n + \kappa}, \quad \kappa > 0. \end{aligned}$$

It is evident to the reader that the Dunkl kernel E*κ*(*iλx*) coincides with the exponential function when the parameter *κ* is equal to zero, i.e., E0(*iλx*) = *eiλx*. This function also has a close connection with the Wright function.

$$\mathcal{E}\_{\mathbf{x}}(\mathbf{x}) = \Gamma(\kappa + 1/2) \left[ \mathcal{W}\_{1,\kappa + 1/2}(\frac{\mathbf{x}^2}{4}) + \frac{\mathbf{x}}{2} \mathcal{W}\_{1,\kappa + 3/2}(\frac{\mathbf{x}^2}{4}) \right],\tag{6}$$

where the Wright function is defined by the series representation, valid in the whole complex plane [26]

$$\mathcal{W}\_{a,\beta}(z) := \sum\_{n=0}^{\infty} \frac{z^n}{n!\Gamma(an+\beta)}, \quad a > -1, \quad \beta \in \mathbb{C}.\tag{7}$$

The Wright function provides a powerful tool for dealing with fractional calculus problems, as it allows for the analysis of fractional differential and integral equations in a unified framework, see [26,27].

The function E*k*(*iξx*) satisfies the following eigenvalue problem

$$\mathcal{O}\_{\mathbb{X}}(\mathcal{E}\_{k}(i\sharp\mathfrak{x})) = i\mathfrak{f}\mathcal{A}\_{\mathbb{X}}^{\diamond}(i\sharp\mathfrak{x}), \quad \mathcal{E}\_{k}(0)) = 1 \tag{8}$$

and has the *Laplace* representation

$$\mathcal{A}\_{\mathbf{x}}(i\mathbf{x}) = \frac{\Gamma(\kappa + 1/2)}{\Gamma(1/2)\Gamma(\kappa)} \int\_{-1}^{1} e^{t\mathbf{x}} (1 - t)^{\kappa - 1} (1 + t)^{\kappa} \, dt. \tag{9}$$

The Dunkl transform is defined by [1,3,4]

$$(\mathcal{J}\_{\mathbf{x}}f)(\boldsymbol{\lambda}) := \int\_{-\infty}^{\infty} f(\mathbf{x}) \, \mathcal{E}\_{\mathbf{x}}(-i\boldsymbol{\lambda}\boldsymbol{x}) \, \sigma\_{\mathbf{x}}(d\boldsymbol{x}).\tag{10}$$

The *Dunkl* transform can be extended to an isometry of *L*2 *κ*(R), that is,

$$\int\_{\mathbb{R}} |f(\mathbf{x})|^2 \, \sigma\_{\mathbf{x}}(d\mathbf{x}) = \int\_{\mathbb{R}} |\widehat{f}\_{\mathbf{x}}(\lambda)|^2 \, \sigma\_{\mathbf{x}}(d\lambda). \tag{11}$$

For any *f* ∈ *L*1 *κ*(R) ∩ *L*2 *κ*(R), the inverse is given by

$$f(\mathbf{x}) = \int\_{\mathbb{R}} \widehat{f}\_{\mathbf{x}}(\lambda) \, \mathcal{E}\_{\mathbf{x}}(i\lambda \mathbf{x}) \, \sigma\_{\mathbf{x}}(d\lambda). \tag{12}$$

As in the classical case, a generalized translation operator was defined in the Dunkl setting side on *L*2 *κ*(R) by Trimèche [3]

$$\mathcal{F}\_{\mathbb{X}}\{\tau^{y}f(\mathbf{x});\xi\} := \mathcal{E}\_{\mathbb{X}}(i\xi y)\mathcal{F}\_{\mathbb{X}}\{f(\mathbf{x});\xi\}, \quad y,\,\xi \in \mathbb{R}.\tag{13}$$

We also define the Dunkl convolution product for suitable functions *f* and *g* by

$$f \ast g(\mathfrak{x}) = \int\_{\mathbb{R}} \pi^{-\mathfrak{x}} f(y) g(y) \sigma\_{\mathfrak{k}}(dy).$$

Explicitly, the generalized translation *τx f*(*y*) takes the explicit form (see [28] Theorem 6.3.7):

$$\begin{split} \tau^{\mathbf{x}} f(y) := & \frac{1}{2} \int\_{-1}^{1} f(\sqrt{\mathbf{x}^2 + y^2 - 2\mathbf{x}yt}) (1 + \frac{\mathbf{x} - y}{\sqrt{\mathbf{x}^2 + y^2 - 2\mathbf{x}yt}}) h\_k(t) dt \\ &+ \quad \frac{1}{2} \int\_{-1}^{1} f(-\sqrt{\mathbf{x}^2 + y^2 - 2\mathbf{x}yt}) (1 - \frac{\mathbf{x} - y}{\sqrt{\mathbf{x}^2 + y^2 - 2\mathbf{x}yt}}) h\_k(t) \, dt \, \end{split} \tag{14}$$

where

$$h\_{\kappa}(t) = \frac{\Gamma(\kappa + 1/2)}{2^{2\kappa}\sqrt{\pi}\Gamma(\kappa)}(1+t)(1-t^2)^{\kappa-1}.$$

#### *2.2. The Generalized Lizorkin Space*

For a comprehensive treatment of the standard Lizorkin space, we recommend referring to the book [12] §2, where the authors provide a detailed and in-depth analysis of this topic. Additionally, the study of the generalized Lizorkin space has been carried out by Soltani [24]. While we cannot provide a detailed overview of the entire subject here, we can highlight some important points for clarity.

We denote by *S*(R) the Schwartz space, which is the space of *C*∞-functions on R which are rapidly decreasing as well as their derivatives, endowed with the topology defined by the seminorms

$$||f||\_{n,m} = \sup\_{\mathbf{x}\in\mathbb{R}, j\le m} (1+\mathbf{x}^2)^n \mathcal{O}\_{\mathbf{x}}^j \boldsymbol{\varrho}(\mathbf{x}), \quad n, m \in \mathbb{N}\_+$$

It is not difficult to check that

$$\mathcal{Q}f(\mathfrak{x}) = f'(\mathfrak{x}) + \kappa \int\_{-1}^{1} f'(\mathfrak{x}t)dt.$$

From this representation, we see that the operator D leaves *S*(R) invariant.

In the context of distribution theory, the space *S* (R) denotes the topological dual of *S*(R), which consists of generalized functions, also known as tempered distributions. The value of a generalized function *f* as a functional on a test function *ϕ* ∈ *S*(R) is denoted by (*f* , *ϕ*).

A generalized function is said to be *κ*-regular if there exists a locally integrable function *f* with respect to the measure *σκ*(*dx*), such that the integral R *f*(*x*)*ϕ*(*x*)*σκ*(*dx*) is finite for every *ϕ* ∈ *S*(R). The action of the *κ*-regular generalized function *f* on a test function *ϕ* is denoted as (*f* , *ϕ*) or equivalently *f* , *ϕκ*. Here, the integral on the right-hand side of the equation is denoted by *f* , *ϕκ*. It is important to note that the measure *σκ*(*dx*) depends on the specific context and properties of the Dunkl operators. The notation and definitions provided above establish a general framework for understanding *κ*-regular generalized functions and their evaluation on test functions.

The Dunkl transform is a powerful mathematical tool that acts as a topological isomorphism between the Schwartz space *S*(R) and itself. This transform extends naturally to generalized functions by considering the Dunkl transform of a generalized function *f* ∈ *S* (R). The definition of the Dunkl transform for generalized functions can be expressed using duality as follows: for any *ϕ* ∈ *S*(R), the pairing between the Dunkl transform of *f* and *ϕ* is given by

$$(\mathcal{J}\_{\mathbb{k}}f, \, \boldsymbol{\varrho}) = (f, \, \mathcal{J}\_{\mathbb{k}}\boldsymbol{\varrho}), \quad \boldsymbol{\varrho} \in \mathcal{S}(\mathbb{R}).$$

In terms of integral notation, it can be written as:

$$\int\_{\mathbb{R}} (\mathcal{A}\_{\mathbf{x}} f)(\mathbf{x}) \varrho(\mathbf{x}) \sigma \kappa(d\mathbf{x}) = \int\_{\mathbb{R}} f(\mathbf{x}) \mathcal{P}\_{\mathbf{x}} \varrho(\mathbf{x}) \sigma \kappa(d\mathbf{x}), \quad \varrho \in \mathcal{S}(\mathbb{R}), \tag{15}$$

provided *f* and F*κ f* are *κ*-regular.

The space *S*(R) itself is not invariant under multiplication by power functions. However, we can define an invariant subspace by utilizing the Dunkl transforms. This leads us to the set Ψ*κ*(R) consisting of functions *ϕ* ∈ *S*(R) that satisfy the conditions:

$$
\partial\_\mathbf{x}^n \varphi(0) = 0, \quad \text{for } n = 0, 1, 2, \dots, n
$$

where D*n κ ϕ* denotes the *n*th order Dunkl transform of *ϕ*. In other words, *ϕ* belongs to Ψ*κ*(R) if all the Dunkl transforms of *ϕ* evaluated at the origin are zero. By imposing these conditions, we construct a space of functions that possess certain transformation properties with respect to the Dunkl operators. The generalized Lizorkin space Φ*κ*(R) is introduced as the *Dunkl* transform preimage of the space Ψ*κ*(R) in the space *S*(R),

$$\Phi\_{\mathbb{K}}(\mathbb{R}) = \left\{ \boldsymbol{\varrho} \in \mathcal{S}(\mathbb{R}) \; : \; \boldsymbol{\varrho} = \mathcal{F}\_{\mathbb{K}}(\boldsymbol{\psi}), \; \boldsymbol{\Psi} \in \Psi\_{\mathbb{K}}(\mathbb{R}) \right\}. \tag{16}$$

According to this definition, any function *ϕ* ∈ Φ*κ*(R) satisfies the orthogonality conditions

$$\int\_{\mathbb{R}} \mathbf{x}^n \, \boldsymbol{\varphi}(\mathbf{x}) \sigma\_\mathbf{x}(d\mathbf{x}) = 0, \quad n = 0, 1, 2, \dots \tag{17}$$

#### **3. Regularization of Integrals with Power Singularity**

In this section, we examine two types of power functions defined on the entire real line

$$\text{sgn}(\mathfrak{x}) := \begin{cases} 1, & \text{if} \quad \mathfrak{x} > 0 \\ -1, & \text{if} \quad \mathfrak{x} < 0. \end{cases}$$

Other types of tempered power functions can be defined as follows

$$\begin{aligned} \mathfrak{x}\_{\pm}^{\mathfrak{a}} &= \frac{1}{2} \left[ |\mathfrak{x}|^{\mathfrak{a}} \pm |\mathfrak{x}|^{\mathfrak{a}} \text{sgn}(\mathfrak{x}) \right], \\ (\pm i \,\mathfrak{x})^{\mathfrak{a}} &= |\mathfrak{x}|^{\mathfrak{a}} \Big( \cos(\pi \mathfrak{a}/2) \pm i \operatorname{sgn}(\mathfrak{x}) \sin(\pi \mathfrak{a}/2) \Big). \end{aligned}$$

These tempered power functions capture different aspects of fractional calculus and are used to generalize the concept of differentiation and integration to noninteger orders.

#### *3.1. Taylor–Dunkl Formula*

To facilitate the forthcoming discussion on analytic continuation, we begin by presenting an additional formula that proves to be valuable in the process.

Let *f* ∈ *C*∞(R); for every *n* ∈ N, we have [19]

$$\tau^y f(\mathbf{x}) = \sum\_{j=0}^{n-1} b\_j(\mathbf{x}) \, \partial\_\mathbf{x}^j f(\mathbf{x}) + r\_n(\mathbf{x}, y; f), \tag{18}$$

where

$$\begin{cases} \begin{aligned} r\_{j+1}(\boldsymbol{x}, \boldsymbol{y}; \boldsymbol{f}) &= \int\_{-|\boldsymbol{y}|}^{|\boldsymbol{y}|} \left( \frac{\text{sgn}(\boldsymbol{y})}{2|\boldsymbol{y}|^{2\kappa}} + \frac{\text{sgn}(\boldsymbol{u})}{2|\boldsymbol{u}|^{2\kappa}} \right) r\_{j}(\boldsymbol{x}, \boldsymbol{u}; \boldsymbol{\mathcal{O}}\_{\mathbf{f}} \boldsymbol{f}) \, |\boldsymbol{u}|^{2\kappa} d\boldsymbol{u}, \\\\ r\_{1}(\boldsymbol{x}, \boldsymbol{y}; \boldsymbol{f}) &= \boldsymbol{\tau}^{y} f(\boldsymbol{x}) - f(\boldsymbol{x}) \end{aligned} \end{cases}$$

and

$$b\_{j+1}(\mathbf{x}) = \int\_{-|y|}^{|y|} (\frac{\text{sgn}(y)}{2|y|^{2\kappa}} + \frac{\text{sgn}(u)}{2|u|^{2\kappa}}) b\_j(u) \, |u|^{2\kappa} \, du, \quad b\_0(\mathbf{x}) = 1. \tag{19}$$

Then,

$$b\_{2s}(\mathbf{x}) = \frac{\Gamma(\kappa + 1/2)}{\Gamma(\kappa + s + 1/2)} \frac{\mathbf{x}^{2s}}{\mathbf{s}!}, \quad b\_{2s+1}(\mathbf{x}) = \frac{\Gamma(\kappa + 1/2)}{\Gamma(\kappa + s + 3/2)} \frac{\mathbf{x}^{2s+1}}{\mathbf{s}!}, \mathbf{s} = 0, 1, 2, \dots$$

From the work of Mourou [23], we can extract the following proposition, which provides a complete asymptotic expansion for *τκ f*(*x*) as *x* approaches *a*.

**Lemma 1.** *Let f* ∈ *C*∞(R) *and a* ∈ R*; then, one has*

$$\pi^a\_\kappa f(\mathbf{x}) \sim \sum\_{s=0}^\infty b\_s(\mathbf{x}) \, \mathcal{O}^s\_\kappa f(a), \quad \text{as} \quad \mathbf{x} \to a,\tag{20}$$

#### *3.2. Generalized Power Functions*

By considering |*x*| −*α* and sign(*x*)|*x*| −*α* as elements of Ψ *κ*(R), we recognize them as *κ*-regular generalized functions for all *α* ∈ C, that is,

$$
\langle |\boldsymbol{x}|^{-a} \boldsymbol{\omega} \cdot \boldsymbol{\varrho} \rangle\_{\mathbf{x}} = \int\_{\mathbb{R}} \frac{1}{|\boldsymbol{\omega}|^{a}} \boldsymbol{\varrho}(\boldsymbol{x}) \boldsymbol{\sigma}\_{\mathbf{x}}(d\boldsymbol{x}) \,. \tag{21}
$$

$$
\langle \text{sgn}(\mathfrak{x}) | \mathfrak{x} |^{-\alpha}, \,\, \phi \rangle\_{\mathbb{K}} = \int\_{\mathbb{R}} \frac{\text{sgn}(\mathfrak{x})}{|\mathfrak{x}|^{\alpha}} \,\, \mathfrak{p}(\mathfrak{x}) \, \sigma\_{\mathfrak{x}}(d\mathfrak{x}).\tag{22}
$$

When considering the functions |*x*| −*α* and sign(*x*)|*x*| −*α* as elements of *S* (R) or Φ *κ*(R), they are not *κ*-regular if (*α*) ≥ 2*κ* + 1. To handle these generalized functions, let *α* ∈ C such that *α* = 2*κ* + 2*s* + 1 for *s* = 0, 1, 2, ... . For *ϕ* ∈ *S*(R), we can define the generalized power function |*x*| −*α* as follows:

$$\begin{split} \left( |\boldsymbol{x}|^{-\mathfrak{a}}, \boldsymbol{\varrho} \right) &= \int\_{|\boldsymbol{x}| < 1} \frac{1}{|\boldsymbol{x}|^{\mathfrak{a}}} \Big[ \boldsymbol{\varrho}(\boldsymbol{x}) - \sum\_{s=0}^{m} b\_{\mathfrak{s}}(\boldsymbol{x}) \, \partial\_{\mathfrak{x}}^{s} \boldsymbol{\varrho}(\boldsymbol{0}) \Big] \boldsymbol{\sigma}\_{\mathfrak{k}}(d\boldsymbol{x}) \\ &+ \sum\_{s=0}^{\left[ \frac{m}{2} \right]} \frac{\partial\_{\mathfrak{k}}^{2s} \boldsymbol{\varrho}(\boldsymbol{0})}{2^{\mathfrak{k}-1/2} \Gamma(\mathfrak{k} + s + 1/2), s!} \frac{1}{2\kappa + 2s + 1 - \mathfrak{a}} \\ &+ \int\_{|\boldsymbol{x}| \ge 1} \frac{\boldsymbol{\varrho}(\boldsymbol{x})}{|\boldsymbol{x}|^{\mathfrak{a}}} \boldsymbol{\sigma}\_{\mathfrak{k}}(d\boldsymbol{x}), \end{split} \tag{23}$$

where *m* > Re(*α*) − 2*κ* − 1. It is important to note that the right-hand side of Equation (23) does not depend on the choice of *m* as long as *m* > (*α*) − 2*κ* − 1. Since *ϕ* ∈ *S*(R), Lemma 1 guarantees that

$$
\rho(\mathfrak{x}) - \sum\_{s=0}^{m} b\_s(\mathfrak{x}) \, \mathcal{Q}\_\kappa^s \rho(0) = \mathcal{O}(\mathfrak{x}^{m+1}) \quad \text{(as} \quad \mathfrak{x} \to 0).
$$

This property ensures the well-definedness of the expression. The mapping *α* → (|*x*| −*α*, *ϕ*) from C to *S* (R) can be extended to a holomorphic function on C − {2*κ* + 2*s* + 1 : *s* = 0, 1, 2, . . . }, with simple poles at *α* = 2*κ* + 2*s* + 1. The residues of the function at these poles are given by

$$\operatorname{Res}\left(\left(|x|^{-\kappa},\,\,\rho\right);2\kappa+2s+1\right) = -\frac{2^{-\kappa+1/2}\partial\_{\kappa}^{2s}\varphi(0)}{\Gamma(\kappa+s+1/2)\,s!}.\tag{24}$$

When *α* = 2*κ* + 2*s* + 1 with *s* = 0, 1, 2, ... , we define the even, tempered power function |*x*| −2*κ*−2*s*−1 as

$$\left( |\mathbf{x}|^{-2\mathbf{x}-2\mathbf{s}-1}, \boldsymbol{\varrho} \right) = \lim\_{\mathbf{a} \to 2\mathbf{x} + 2\mathbf{s} + 1} \left\{ \left( |\mathbf{x}|^{-\mathbf{a}}, \boldsymbol{\varrho} \right) + \frac{\partial\_{\mathbf{x}}^{2\mathbf{s}} \boldsymbol{\varrho}(0)}{2^{\mathbf{x} - 1/2} \Gamma(\mathbf{x} + \mathbf{s} + 1/2) \mathbf{s}!} \frac{1}{\mathbf{a} - 2\mathbf{x} - 2\mathbf{n} - 1} \right\}. \tag{25}$$

This provides a definition for the even, tempered power |*x*| −*α* for all *α* ∈ C.

Similarly, for *α* ∈ C such that *α* = 2*κ* + 2*s* + 2 with *s* = 0, 1, 2 ... , we define the odd tempered power function |*x*| −*α*sgn(*x*) by

$$\begin{split} \left(\frac{\text{sgn}(\mathbf{x})}{|\mathbf{x}|^{\alpha}},\,\boldsymbol{\varrho}\right) &= \int\_{|\mathbf{x}|<1} \frac{\text{sgn}(\mathbf{x})}{|\mathbf{x}|^{\alpha}} \left[\boldsymbol{\varrho}(\mathbf{x}) - \sum\_{s=0}^{m} b\_{\mathbf{s}}(\mathbf{x}) \mathcal{Q}\_{\mathbf{x}}^{s} \boldsymbol{\varrho}(\mathbf{0})\right] \boldsymbol{\sigma}\_{\mathbf{x}}(d\mathbf{x}) \\ &+ \sum\_{s=0}^{\lfloor\frac{m-1}{2}\rfloor} \frac{\mathcal{Q}\_{\mathbf{x}}^{2s+1} \boldsymbol{\varrho}(\mathbf{0})}{2^{\kappa-1/2} \Gamma(\kappa+s+3/2) \, s!} \frac{1}{2\kappa+2s+2-a} \\ &+ \int\_{|\mathbf{x}|\ge 1} \frac{\text{sgn}(\mathbf{x})}{|\mathbf{x}|^{\alpha}} \boldsymbol{\varrho}(\mathbf{x}) \, \boldsymbol{\sigma}\_{\mathbf{x}}(d\mathbf{x}) \quad (m > \Re(\alpha) - 2\kappa - 2). \end{split} \tag{26}$$

It follows that the mapping *α* → (|*x*| −*α*sgn(*x*), *ϕ*) is analytic on C − {2*κ* + 2*s* + 2, s = 0, 1, 2, . . . }, with simple poles at *α* = 2*κ* + 2*s* + 2 and

$$\operatorname{Res}\left(\left(|x|^{-\mathfrak{a}}\operatorname{sgn}(x),\,\varrho\right);2\kappa+2s+2\right) = -\frac{2^{-\kappa+1/2}\partial\_{\mathfrak{k}}^{2s+1}\varrho(0)}{\Gamma(\kappa+s+3/2)s!}.$$

For *α* = 2*κ* + 2*s* + 2, with *s* = 0, 1, 2, ... , we define the odd, tempered powers function sgn(*x*)|*x*| −2*κ*−2*s*−2 as

$$\delta(\operatorname{sgn}(\mathbf{x})|\mathbf{x}|^{-2\mathbf{x}-2\mathbf{s}-2},\boldsymbol{\varrho}) = \lim\_{\mathbf{z}\to\mathbf{2x}+2\mathbf{s}+\mathbf{2}} \left\{ (\operatorname{sgn}(\mathbf{x})|\mathbf{x}|^{-\mathbf{a}},\boldsymbol{\varrho}) + \frac{\beta\_{\mathbf{x}}^{2\mathbf{s}+1}\boldsymbol{\varrho}(0)}{2^{\mathbf{s}-1/2}\Gamma(\mathbf{x}+\mathbf{s}+3/2)\mathbf{s}!} \frac{1}{\mathbf{a}-2\mathbf{s}-2\mathbf{s}-2} \right\}.\tag{27}$$

#### **4. Fractional-Type Integral and Derivative for the Dunkl Operator**

In this section, we embark on a comprehensive exploration of fractional-type integral operators associated with the Dunkl operator. These operators transcend the conventional Riesz fractional integral, Feller fractional integral, and Liouville fractional integral, as they are specifically designed to operate within the Dunkl setting.

#### *4.1. The Riesz–Dunkl Fractional Integral*

In this section, our focus lies on extending the Riesz fractional integral to any arbitrary value of (*α*) > 0. As a reminder, the Riesz fractional integral *Iα f* is defined by

$$(I^\mathfrak{a}f)(\mathfrak{x}) = \frac{1}{\gamma(\mathfrak{a})} \int\_{\mathbb{R}} k\_\mathfrak{a}(\mathfrak{x} - y) f(y) dy. \tag{28}$$

where *kα*(*x*) is defined as:

$$k\_a(\mathbf{x}) = \begin{cases} |\mathbf{x}|^{\alpha - 1}, & \mathfrak{a} \neq 1, \mathfrak{z}, \mathfrak{z}, \dots, \\ -|\mathbf{x}|^{\alpha - 1} \ln |\mathbf{x}|, & \mathfrak{a} = 1, \mathfrak{z}, \mathfrak{z}, \dots \end{cases} \tag{29}$$

The normalization factor *γ*(*α*) depends on the value of *α* and is given by:

$$\gamma(\mathfrak{a}) = \begin{cases} \frac{2^{a-1/2}\pi^{1/2}\Gamma(\frac{\mathfrak{a}}{2})}{\Gamma(\frac{1-\mathfrak{a}}{2})}, & \mathfrak{a} \neq 2\mathfrak{s}+1, \; \mathfrak{s} = 0, 2, \dots, \\\ (-1)^{\mathfrak{s}} \mathfrak{s}! \pi^{1/2} 2^{2\mathfrak{s}} \Gamma(\mathfrak{s}+1/2), & \mathfrak{a} = 2\mathfrak{s}+1, \; \mathfrak{s} = 0, 2, \dots \end{cases} \tag{30}$$

**Lemma 2.** *Let κ* < *α* < 2*κ* + 1*. Then, the Dunkl transform of* |*x*| *α*−2*κ*−1 *exists in the usual sense, and it is given by*

$$\, \mathcal{P}\_{\kappa}^{-1}(|\boldsymbol{x}|^{-\alpha}) = \frac{\Gamma(\kappa + \frac{1-\alpha}{2})}{2^{\kappa - \kappa - 1/2}\Gamma(\frac{\alpha}{2})} |\boldsymbol{x}|^{\kappa - 2\kappa - 1}.$$

**Proof.** By using (5), we obtain

$$\begin{split} \mathcal{R}\_{\mathbb{X}}^{-1} \left( |\mathbf{x}|^{-\alpha} \right) (\mathbf{x}) &= \int\_{-\infty}^{\infty} |u|^{-\alpha} \mathcal{E}\_{\mathbb{X}} (i u \boldsymbol{\omega}) \sigma\_{\mathbb{X}} (du) \\ &= \frac{2}{2^{\kappa + 1/2} \Gamma(\kappa + \frac{1}{2})} \int\_{0}^{\infty} \mathcal{J}\_{\mathbb{X} - 1/2} (|\mathbf{x}| u) u^{-\kappa + 2\kappa} \, du. \end{split}$$

Making the substitution *t* = |*x*|*u* yields

$$\,\_3\mathcal{F}\_{\kappa}^{-1}(|\mathfrak{x}|^{-a})(\mathfrak{x}) = |\mathfrak{x}|^{\alpha - 2\kappa - 1} \int\_0^\infty \frac{f\_{\kappa - 1/2}(u)}{u^{\alpha - \kappa - 1/2}} \, du \, \mathfrak{x}$$

The result follows from the following Weber formula [29] §13.24:

$$\int\_0^\infty \frac{f\_\nu(t)}{t^{\nu-\mu+1}} \, dt = \frac{1}{2^{\nu-\mu+1}} \frac{\Gamma(\frac{\mu}{2})}{\Gamma(\nu-\frac{\mu}{2}+1)}, \quad 0 < \Re(\mu) < \Re(\nu) + \frac{3}{2}. \tag{31}$$

**Proposition 1.** *The Dunkl transform of* |*x*| −*α* ∈ Ψ *κ*(R) *is given by*

$$\mathbb{P}\_{\mathbb{X}}\mathbb{P}\_{\mathbb{X}}^{-1}\left(|\mathbb{x}|^{-a}\right) = \frac{1}{\gamma\_{\mathbb{X}}(a)} \begin{cases} |\mathbb{x}|^{a-2\kappa-1}, & a \neq -2s, \ a \neq 2\kappa+2s+1, s \in \mathbb{N}\_{0}, \\\\ |\mathbb{x}|^{a-2\kappa-1} \ln \frac{1}{|\mathbb{x}|}, & a = 2\kappa+2s+1, s \in \mathbb{N}\_{0}, \\\\ (-1)^{s}\partial\_{\mathbb{X}}^{2s}\delta, & a = -2s, s \in \mathbb{N}\_{0}. \end{cases}$$

*where*

$$\gamma\_{\mathbb{K}}(\mathfrak{a}) = \begin{cases} \frac{2^{\mathfrak{a}-\mathfrak{x}-1/2}\Gamma(\frac{\mathfrak{a}}{2})}{\Gamma(\mathfrak{x}+\frac{1-\mathfrak{a}}{2})} & \mathfrak{a} \neq -2\mathfrak{s}, \ \mathfrak{a} \neq 2\mathfrak{x} + 2\mathfrak{s} + 1, \\\\ (-1)^{\mathfrak{s}} \mathfrak{s}! 2^{\mathfrak{x}+2\mathfrak{s}+1/2} \Gamma(\mathfrak{x}+\mathfrak{s}+1/2), & \mathfrak{a} = 2\mathfrak{x} + 2\mathfrak{s} + 1, \\\\ 1, & \mathfrak{a} = -2\mathfrak{s} \end{cases}$$

*and δ is the Dirac delta distribution.*

**Proof.** From Lemma 2, it is evident that by analytic continuation, for *α* ∈ C such that *α* = 2*κ* + 2*s* + 1 and *α* = −2*s* for *s* = 0, 1, 2, . . . , we have:

$$\frac{1}{|\boldsymbol{\chi}|^{\alpha}} = \frac{\Gamma(\kappa + \frac{1-\alpha}{2})}{2^{\kappa-\kappa-1/2}\Gamma(\frac{\kappa}{2})} \cdot \mathcal{P}\_{\mathbb{K}}(|\boldsymbol{\chi}|^{\alpha-2\kappa-1}).\tag{32}$$

The case *α* = −2*s* for *s* = 0, 1, 2, . . . follows from the fact that

$$(\mathcal{P}\_\kappa \partial\_\kappa^{2s} \varrho)(\mathfrak{x}) = (-1)^s |\mathfrak{x}|^{2s} (\mathcal{P}\_\kappa \varrho)(\mathfrak{x}), \quad \varrho \in \mathcal{S}(\mathbb{R}).$$

It remains to consider the case *α* = *αs* = 2*κ* + 2*s* + 1 for *s* ∈ N0. From Equation (32), we have

$$\frac{\partial}{\partial a}\left((a-a\_s)(|x|^{-a},\mathcal{P}\_\mathbf{k}\boldsymbol{\varrho})\right) = \frac{\partial}{\partial a}\left(\eta\left(a\right)(|x|^{a-2\kappa-1},\boldsymbol{\varrho})\right), \quad \eta\left(a\right) = \frac{\kappa-a\_s}{\gamma\_\kappa(a)}.\tag{33}$$

By considering (23) and (25), the limit as *α* → *αs* of the left-hand side of (33) can be evaluated as follows:

$$\lim\_{\alpha \to \alpha\_\delta} \frac{\partial}{\partial \alpha} \left( (\alpha - \alpha\_k) (|x|^{-\alpha}, \mathcal{P}\_k \boldsymbol{\varphi}) \right) = (|x|^{-2\kappa - 2s - 1}, \mathcal{P}\_k \boldsymbol{\varphi}).$$

The limit of the right-hand side of Equation (33) as *α* → *αs* can be evaluated as follows:

$$\lim\_{a \to a\_\delta} \frac{\partial}{\partial a} \left( \eta(a) \left( |x|^{a-2\kappa-2}, \,\, \phi \right) \right) = \lim\_{a \to a\_\delta} \left( (\eta'(a) + \eta(a) \ln|x|) |x|^{a-2\kappa-1}, \,\, \phi \right).$$

A straightforward computation shows that

$$\lim\_{a \to a\_s} \eta(a) = \frac{(-1)^{s+1}}{s! 2^{\kappa + 2s - 1/2} \Gamma(\kappa + s + 1/2)}.\tag{34}$$

Taking into account Equation (17), in the limit as *α* approaches *αs*, we obtain the following expression:

$$\Gamma\left(|\mathbf{x}|^{-2\mathbf{x}-2s-1}, \mathcal{F}\_{\mathbf{x}}\boldsymbol{\varrho}\right) = \frac{(-1)^{s}}{\mathrm{s!}2^{\mathbf{x}+2s-1/2}\Gamma(\mathbf{x}+\mathbf{s}+1/2)}(|\mathbf{x}|^{2s}\ln\frac{1}{|\mathbf{x}|}\; \boldsymbol{\varrho}).\tag{35}$$

**Definition 1.** *For* (*α*) > 0, *we define the Riesz–Dunkl fractional integral* I *α κ f of f* ∈ Φ*κ*(R) *as:*

$$(\mathcal{J}\_\mathbf{x}^\mathbf{a} f)(\mathbf{x}) = \int\_{\mathbb{R}} \pi^{-y} \mathcal{X}\_{\mathbf{x},\mathbf{a}}(\mathbf{x}) f(y) \sigma\_\mathbf{x}(dy) \tag{36}$$

*where*

$$\mathcal{K}\_{\mathbf{x},\mathbf{a}}(\mathbf{x}) = \frac{1}{\gamma\_{\mathbf{x}}(\mathbf{a})} \begin{cases} |\mathbf{x}|^{\mathbf{a} - 2\mathbf{x} - 1}, & \mathbf{a} \neq -2\mathbf{s}, \; \mathbf{a} \neq 2\mathbf{x} + 2\mathbf{s} + 1 \\\\ \ln(\frac{1}{|\mathbf{x}|}) |\mathbf{x}|^{\mathbf{a} - 2\mathbf{x} - 1}, & \mathbf{a} = 2\mathbf{x} + 2\mathbf{s} + 1. \end{cases} \tag{37}$$

The following theorem states that the space Φ*κ*(R) is closed under the action of the operator I *α κ* . This result ensures the consistency and coherence of the space Φ*κ*(R) under the Riesz–Dunkl fractional integral. Moreover, the proposition establishes the relationship between the Dunkl transform F*κ* and the fractional integral operator I *α κ* and shows the compatibility of the fractional integral operators I *α κ* under composition.

**Theorem 1.** *The space* Φ*κ*(R) *is invariant under the operator* I *α κ , i.e.,*

$$f \in \Phi\_{\mathbb{K}}(\mathbb{R}) \quad \Rightarrow \quad \mathcal{J}\_{\mathbb{K}}^{\alpha}f \in \Phi\_{\mathbb{K}}(\mathbb{R}).$$

*Furthermore,*

$$\text{The first-order coupling between the two-dimensional } \mathcal{N} \text{-matrices is the only possible } \mathcal{N} \text{-matrices with } \mathcal{N} = \{0, 1, 2, \dots, N\} \text{ and } \mathcal{N} = \{0, 1, 2, \dots, N\}.$$

*and*

$$\left(\mathcal{J}\_{\mathbb{K}}\mathcal{J}\_{\mathbb{K}}^{\alpha}f\right) = \frac{1}{|\mathfrak{x}|^{\alpha}}\mathcal{J}\_{\mathbb{K}}f$$

I *α κ* I *β κ* = I *α*+*β κ* , (*α*), (*δ*) > 0.

The proof of this theorem is omitted, but it can be established by utilizing Lemma 2 and Proposition 1 mentioned earlier, which provide the necessary tools and results to derive these conclusions.

Utilizing the reflection formula for the gamma function, we have:

$$
\Gamma(z)\Gamma(1-z) = \frac{\pi z}{\sin \pi z}, \quad z \notin \mathbb{Z}.
$$

In the limit when *κ* ↓ 0, we retrieve the classical *Riesz* and *Feller fractional integral* (see, [11]) §12.1

$$\lim\_{\mathbf{x}\downarrow 0} \mathcal{J}\_{\mathbf{x}}^{a} f(\mathbf{x}) = \frac{1}{2\Gamma(a)\cos(\pi a/2)} \int\_{-\infty}^{\infty} \frac{1}{|\mathbf{x}-\mathbf{y}|^{1-a}} f(\mathbf{y}) d\mathbf{y}.\tag{38}$$

#### *4.2. Feller–Dunkl Fractional Integral*

In this section, we aim to establish an analogous version of the classical Feller fractional integral within the framework of Dunkl operators. The Feller fractional integral, denoted as *Jα κ f*(*x*), is defined as follows:

$$J\_{\kappa}^{a}f(\mathbf{x}) = \frac{1}{2\Gamma(\alpha)\sin(\pi\alpha/2)} \int\_{-\infty}^{\infty} \frac{\text{sgn}(\mathbf{x}-y)}{|\mathbf{x}-y|^{1-a}} f(y) dy. \tag{39}$$

The following lemmas play a crucial role in establishing an extension of the Feller integral within the framework of the Dunkl operator.

**Lemma 3.** *Let κ* < *α* < 2*κ* + 2*. Then, the Dunkl transform of sgn*(*x*)|*x*| −*α exists in the usual sense, and it is given by*

$$\,\_3\mathcal{F}\_\kappa^{-1}(\operatorname{sgn}(\mathbf{x})|\mathbf{x}|^{-\alpha}) = i \frac{\Gamma(\kappa + \frac{2-\alpha}{2})}{2^{\alpha-\kappa-1/2}\Gamma(\frac{1+\alpha}{2})} \operatorname{sgn}(\mathbf{x}) \, |\mathbf{x}|^{\alpha-2\kappa-1}.$$

**Proof.** Using (5), we have

$$\begin{split} \mathcal{F}\_{\mathbf{x}}^{-1} \left( \operatorname{sgn}(\mathbf{x}) |\mathbf{x}|^{-a} \right) (\mathbf{x}) &= \int\_{-\infty}^{\infty} \operatorname{sgn}(u) |u|^{-a} \mathcal{E}\_{\mathbf{x}} (i \mathbf{x} \mathbf{x}) \sigma\_{\mathbf{x}} (du) \\ &= \frac{2 \operatorname{i} \mathbf{x}}{(2\kappa + 1)2^{\kappa + 1/2} \Gamma(\kappa + \frac{1}{2})} \int\_{0}^{\infty} \mathcal{J}\_{\mathbf{x} + 1/2}(\mathbf{x} u) \, u^{-a + 2\kappa + 1} \, du \\ &= i \operatorname{sgn}(\mathbf{x}) |\mathbf{x}|^{a - 2\kappa - 1} \int\_{0}^{\infty} \frac{I\_{\mathbf{x} + 1/2}(t)}{t^{a - \kappa - 1/2}} \, dt. \end{split}$$

The Weber Formula (31) achieves the result.

**Lemma 4.** *The following holds: for α* = 2*κ* + *s* + 1 *with s* ∈ Z−, *we have*

$$
\left| \mathcal{Q}\_{\mathbf{x}} |\mathbf{x}|^{-\alpha} = -\mathfrak{a} \left| \mathfrak{x} \right|^{-\alpha - 1} \text{sgn}(\mathfrak{x}).
$$

**Proof.** Let *κ* < (*α*) < 2*κ* + 1 and *ϕ* ∈ *S*(R), we have

$$\begin{aligned}<\left.\partial\_{\mathbb{K}}|\boldsymbol{x}|\,^{-\alpha}\right|\,^{\alpha}\boldsymbol{\varrho} &>\_{\mathbb{K}} = - <|\boldsymbol{x}|^{-\alpha}\,,\partial\_{\mathbb{K}}\boldsymbol{\varrho} >\_{\mathbb{K}}\\ &= -\int\_{\mathbb{R}} |\boldsymbol{x}|^{-\alpha}\,\partial\_{\mathbb{K}}\boldsymbol{\varrho}(\boldsymbol{x})\boldsymbol{\sigma}\_{\mathbb{K}}(d\boldsymbol{x})\\ &= -\alpha\int\_{\mathbb{R}} |\boldsymbol{x}|^{-\alpha-1}\text{sgn}(\boldsymbol{x})\boldsymbol{\varrho}(\boldsymbol{x})\boldsymbol{\sigma}\_{\mathbb{K}}(d\boldsymbol{x})\\ &= -\alpha <|\boldsymbol{x}|^{-\alpha-1}\text{sgn}(\boldsymbol{x})\boldsymbol{\varrho} >\_{\mathbb{K}}.\end{aligned}$$

By analytic continuation for *α* ∈ C such that *α* = 2*κ* + *s* + 1, *s* ∈ N, we have

$$\mathcal{Q}\_{\mathbb{X}}|\mathfrak{x}|^{-\mathfrak{a}} = -\mathfrak{a}|\mathfrak{x}|^{-\mathfrak{a}-1}\text{sgn}(\mathfrak{x})\_{\mathfrak{a}}$$

which is the required result.

**Proposition 2.** *The Dunkl transform of sgn*(*x*)|*x*| −*α* ∈ Ψ *κ*(R) *is given by*

$$\mathcal{R}\_{\mathbf{x}}^{-1}\left(-i\,|\mathbf{x}|^{-a}\text{sgn}(\mathbf{x})\right) = \frac{1}{\delta\_{\mathbf{x}}(a)}\begin{cases} \text{sgn}(\mathbf{x})\,|\mathbf{x}|^{a-2\kappa-1}, & a \neq -2s-1, \, a \neq 2\kappa+2s+2, \, s \in \mathbb{N}\_{0}, \\\\ -|\mathbf{x}|^{2s+1}\ln|\mathbf{x}|, & a = 2\kappa+2s+2, \, s \in \mathbb{N}\_{0}, \\\\ (-1)^{s}\partial\_{\kappa}^{2s+1}\delta\_{\times} & \mathfrak{a} = -2s-1, \, s \in \mathbb{N}\_{0}. \end{cases}$$

*where*

$$\delta\_{\mathbf{k}}(\boldsymbol{\alpha}) = \begin{cases} \frac{2^{\kappa - \kappa - 1/2} \Gamma(\frac{\kappa + 1}{2})}{\Gamma(\kappa + \frac{2 - \kappa}{2})} & \boldsymbol{\alpha} \neq -2\boldsymbol{s} - 1, \; \boldsymbol{\alpha} \neq 2\boldsymbol{\kappa} + 2\boldsymbol{s} + 2, \\\\ (-1)^{s} \boldsymbol{s}! 2^{\kappa + 2s + 3/2} \Gamma(\kappa + \boldsymbol{s} + 3/2), & \boldsymbol{\alpha} = 2\boldsymbol{\kappa} + 2\boldsymbol{s} + 2, \\\\ 1, \; \boldsymbol{\alpha} = -2\boldsymbol{s} - 1. \end{cases}$$

**Proof.** The proof of the proposition can be achieved by utilizing the above lemmas.

**Definition 2.** *For* (*α*) > 0*, we define the Riesz–Dunkl fractional integral* J *α κ f of f* ∈ Φ*κ*(R) *as:*

$$(\mathcal{J}\_{\kappa}^{\alpha}f)(\mathbf{x}) = \int\_{\mathbb{R}} \pi^{-y} \mathcal{J}\_{\kappa,\alpha}(\mathbf{x}) f(y) \sigma\_{\mathbf{x}}(dy) \tag{40}$$

*where*

$$\mathcal{H}\_{\mathbf{k},\mathbf{a}}(\mathbf{x}) = \frac{1}{\delta\_{\mathbf{k}}(\mathbf{a})} \begin{cases} \operatorname{sgn}(\mathbf{x}) |\mathbf{x}|^{\mathbf{a} - 2\mathbf{x} - 1}, & \mathbf{a} \neq 2\mathbf{x} + 2\mathbf{s} + 2 \\\\ \operatorname{sgn}(\mathbf{x}) \ln(\frac{1}{|\mathbf{x}|}) |\mathbf{x}|^{\mathbf{a} - 2\mathbf{x} - 1}, & \mathbf{a} = 2\mathbf{x} + 2\mathbf{s} + 2. \end{cases} \tag{41}$$

In the limit when *α* ↓ 0, we obtain

$$\lim\_{\kappa \downarrow 0} (\mathcal{J}\_\kappa^a f) := \mathcal{J}\ell\_\mathbf{k} f(\mathbf{x}) := \frac{\Gamma(\kappa + 1)}{\sqrt{\pi} \Gamma(\kappa + 1/2)} \lim\_{\varepsilon \downarrow 0} \int\_{|y| \ge \varepsilon} \pi\_\mathbf{x}^{-y} f(\mathbf{x}) \frac{dy}{y},\tag{42}$$

and

$$(\mathcal{J}\_\kappa \mathcal{A}\_\kappa^\varrho f)(\mathfrak{x}) = -i \operatorname{sgn}(\mathfrak{x}) (\mathcal{J}\_\kappa f)(\mathfrak{x}), \quad f \in \Phi\_\kappa(\mathbb{R}).$$

For the special case of *κ* = 0 and *α* = 0, the Feller–Dunkl fractional integral coincides with the Hilbert transform. The Hilbert transform is a well-known operator in harmonic analysis and signal processing. It acts as a multiplier with the symbol −*isign*(*x*).

It can be easily seen from Propositions 1 and 2 that the operators I *α κ* and J *α κ* are connected by

> I *α κ* = H*κ*J *α κ* .

#### *4.3. Riemann–Liouville–Dunkl fractional integrals*

The Riemann–Liouville fractional integrals are given by [12] formulas (5.1) and (5.2)

$$I\_+^a f(\mathbf{x}) := \frac{1}{\Gamma(a)} \int\_{-\infty}^{\infty} (\mathbf{x} - \mathbf{y})^{a-1} f(\mathbf{y}) d\mathbf{y} \tag{43}$$

and

$$I\_{-}^{\mathfrak{a}}f(\mathfrak{x}) := \frac{1}{\Gamma(\mathfrak{a})} \int\_{\mathfrak{x}}^{\infty} (y-\mathfrak{x})^{\mathfrak{a}-1} f(y) dy \tag{44}$$

They are related to the Riesz fractional integral *Iα* and its conjugate *Jα* by

$$\begin{aligned} I^\alpha f(\mathbf{x}) &= \frac{I\_+^\alpha f(\mathbf{x}) + I\_-^\alpha f(\mathbf{x})}{2\cos(\frac{\pi\alpha}{2})}, \\\ J^\alpha f(\mathbf{x}) &= \frac{I\_+^\alpha f(\mathbf{x}) - I\_-^\alpha f(\mathbf{x})}{2\sin(\frac{\pi\alpha}{2})}. \end{aligned}$$

Similarly, the correspondent definition of the Riemann–Liouville–Dunkl fractional integral can be given as follows:

$$\begin{cases} \mathcal{J}\_{\kappa,+}^{\mathfrak{a}}f(\mathfrak{x}) := \cos(\alpha \pi/2) \mathcal{J}\_{\kappa}^{\mathfrak{a}}f(\mathfrak{x}) + \sin(\alpha \pi/2 \mathcal{J}\_{\kappa}^{\mathfrak{a}}f(\mathfrak{x}), \\\mathcal{J}\_{\kappa,-}^{\mathfrak{a}}f(\mathfrak{x}) := \cos(\alpha \pi/2) \mathcal{J}\_{\kappa}^{\mathfrak{a}}f(\mathfrak{x}) - \sin(\alpha \pi/2 \mathcal{J}\_{\kappa}^{\mathfrak{a}}f(\mathfrak{x}). \end{cases}$$

**Proposition 3.** *The following holds:*

*(1) For f* ∈ Φ, *we have*

$$(\mathcal{J}\_{\mathbb{X}}\mathcal{J}\_{\kappa,\pm}^{\alpha}f) = (\mp i\mathfrak{x})^{-\mathfrak{a}}\left(\mathcal{J}\_{\mathbb{X}}f\right)(\mathfrak{x}).$$

*(2) For f* ∈ Φ *and* (*α*), (*β*) > 0, *we have*

$$
\mathcal{J}\_{\kappa,\pm}^{\alpha} \mathcal{J}\_{\kappa,\pm}^{\beta} = \mathcal{J}\_{\kappa,\pm}^{\alpha+\beta} \cdot .
$$

*(3) Integration by parts:*

$$\int\_{\mathbb{R}} \mathcal{J}\_{\mathbf{x},+}^{a} f(\mathbf{x}) \, g(\mathbf{x}) \sigma\_{\mathbf{k}}(d\mathbf{x}) = \int\_{\mathbb{R}} f(\mathbf{x}) \, \mathcal{J}\_{\mathbf{x},-}^{a} g(\mathbf{x}) \sigma\_{\mathbf{k}}(d\mathbf{x}), \quad f, \, \mathbf{g} \in \Phi\_{\mathbf{k}}(\mathbb{R}).$$

**Funding:** This research was funded by the Researchers Supporting Project number (RSPD2023R974), King Saud University, Riyadh, Saudi Arabia.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author extends his appreciation to the Researchers Supporting Project number (RSPD2023R974), King Saud University, Riyadh, Saudi Arabia.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.


**13.2 Applications of bone scintigraphy in pediatric populations**

#### **13.2.1 Infections**

Acute osteomyelitis is a common pediatric disease that mostly affects children under 5 years old. It usually is the result of hematogenous spread of infection due to the rich vascular supply of the growing skeleton. Typically, bone scan become positive 24 to 72 hours after the onset of infection, while plain films do not manifest evidence of infection until 3 to 4 weeks after. Therefore, three-phase bone scintigraphy is the most sensitive imaging modality for early diagnosis. The sensitivity of a three-phase bone scan has been estimated as 94% with a specificity of 95% (Shammas, 2009; as cited in Schauwecker, 1992). Ideally, scintigraphic imaging should be obtained before joint aspiration, and a delayed whole-body scan on skeletal phase should be obtained because osteomyelitis in childhood can be multifocal or present with referred pain. In addition, malignant disease such as leukemia

Proper positioning is important in pediatrics particularly in young infants, and although children are smaller, it does not imply that more of a child can be imaged on a single scintigraphic view. In fact, examinations take longer in children and infants because of the requirement of joint-to-joint images for detailed assessment. Although the new gamma camera systems often allow whole-body passes it is often necessary to supplement these images with magnified spot views or even pinhole imaging. Image magnification either with camera zoom, computer magnification, or collimation is essential when performing scintigraphic examinations in children. Magnification is either optical with collimation or electronic. Optical magnification uses either a pinhole or converging collimator, enlarges the image, and improves overall system resolution. Electronic magnification makes the image bigger without altering overall system resolution. The capability for SPECT imaging is essential in pediatric scintigraphy. SPECT allows for improved image contrast and hence improved diagnostic accuracy. It is helpful in localizing and further defining most musculoskeletal abnormalities to include the extremities and is essential when assessing a child with the clinical problem of back pain. Multiple head detector gamma camera systems are becoming more available in pediatric centers. The advantages of these systems include increased resolution and sensitivity and decreased time of examination in a child. Correlative imaging is essential to state of the art practice of pediatric nuclear medicine. Computer multimodality image fusion programs are becoming available and more sophisticated. They allow comparison of different isotope scintigraphic studies or serial studies in the same patient or comparison of scintigraphy with other imaging modalities, such as CT, MR imaging, and PET for better correlation of anatomy and function. New combined gamma camera and CT devices allowing direct anatomic and physiologic correlation are also being manufactured and will have further impact on the care of the

The normal distribution in a pediatric bone scan may differ from adults (Shammas, 2009; as cited in Nadel, 2007). In children there is high physeal and apophyseal uptake due to their rich blood supply and active enchondral ossification. Absence of uptake in nonossified cartilaginous structures should not be mis- taken for avascular necrosis. Regions where this may be of concern in younger children include the femoral capital epiphysis, patella, and navicular bone. Before ossification, the ischiopubic synchondrosis appears as a discontinuity of the inferior pubic ramus. During ossification, increased uptake in ischiopubic synchondroses is a common normal variant and should not be misinterpreted as a

Acute osteomyelitis is a common pediatric disease that mostly affects children under 5 years old. It usually is the result of hematogenous spread of infection due to the rich vascular supply of the growing skeleton. Typically, bone scan become positive 24 to 72 hours after the onset of infection, while plain films do not manifest evidence of infection until 3 to 4 weeks after. Therefore, three-phase bone scintigraphy is the most sensitive imaging modality for early diagnosis. The sensitivity of a three-phase bone scan has been estimated as 94% with a specificity of 95% (Shammas, 2009; as cited in Schauwecker, 1992). Ideally, scintigraphic imaging should be obtained before joint aspiration, and a delayed whole-body scan on skeletal phase should be obtained because osteomyelitis in childhood can be multifocal or present with referred pain. In addition, malignant disease such as leukemia

pediatric patient.

**13.2.1 Infections**

pathological lesion (Shammas, 2009).

**13.2 Applications of bone scintigraphy in pediatric populations**

and sarcoma may mimic acute osteomyelitis (Shammas, 2009; as cited in Connolly et al., 2007; Ma et al., 2007). All three phases of the bone scan show focally high uptake in the affected bone. Occasionally, the affected bone in children shows low uptake or a photopenic defect (cold osteomyelitis) (Shammas, 2009; as cited in Pennington, 1999). This is most likely due to reduced tracer delivery by increased intraosseous pressure from inflammation, oedema, and joint effusion (Shammas, 2009). Cellulitis may be differentiated from osteomyelitis because the former typically demonstrates diffuse increased activity in the soft tissues on the first two phases, without focal osseous abnormality on the third phase (Shammas, 2009; as cited in Wegener & Alavi, 1991). Although chronic recurrent multifocal osteomyelitis (CRMO) and acute osteomyelitis share a common histopathologic feature, namely chronic inflammation, they are different in important ways (Nadel & Stilwell, 2001). CRMO occurs most frequently in the latter half of the first decade and the first half of the second decade of life, and it is more common in girls, differently from acute osteomyelitis that occurs in children under 5 years old (Shammas, 2009). A predisposing cause is not found for CRMO in contrast to conventional osteomyelitis (Nadel & Stilwell, 2001). Bone scintigraphy is helpful in identifying the multifocal bone lesions and characteristically displays high uptake in both symptomatic and asymptomatic lesions (Shammas, 2009, as cited in Connolly, 2007). Other infection typical of children under 3 years of age is septic arthritis. Monoarticular involvement is the most common pattern. The more affected joints are the knees and the hips. As in the osteomyelitis, there is an increased uptake on all three phases of three-phase bone scan, but in septic arthritis there is a symmetric uptake in both sides of the joint (Diaz& De Haro, 2005). Transient synovitis is the most common condition that mimics septic arthritis. In this case, three-phase bone scan may be normal or may show diffuse increased activity on the first two phases. Delayed images may displa periarticular increased activity in the affected joint (Shammas, 2009).


**5. Conclusions**

In this work, we developed a 3D avatar-based sign language learning system that converts the English speech or text into corresponding ISL movements. Initially, the input speech is converted into an equivalent English sentence using the IBM-Watson service. The converted English text is further translated into the corresponding ISL sentence using regular expression and script generator. Finally, each word of the ISL sentence is transformed into an equivalent sign movement, represented by a 3D avatar. The translation method (English–ISL and ISL–sign movement) has been evaluated by the SER, BLEU, and NIST metrics. The proposed model achieved an 82.30% BLEU score that represents

the translation accuracy from English to ISL sentences. The text-to-sign translation model (from ISL sentence to sign movement) achieved a 10.50 SER score, which signifies that 89.50% of signs were correctly generated by the 3D avatar model for the respective ISL sentence. It may be noted that the proposed system has been developed for a limited corpus, and no facial expressions were included, which can be considered as an important part of any sign language system. The transition between the signs while performing a sign sentence can be improved further by learning specific transitions based on hand positions while signing. Moreover, the sign language recognition system that converts a sign to text/speech is significantly more difficult to develop. Such a system can be added within the proposed framework to build a complete sign language interpretation system.

**Author Contributions:** Conceptualization, D.D.C. and P.K.; methodology, D.D.C., P.K. and S.M.; software, S.M.; validation, D.D.C. and P.K.; formal analysis, D.D.C. and P.K.; investigation, P.P.R. and M.I.; data curation, D.D.C. and P.K.; writing—D.D.C., P.K.; writing—review and editing, P.P.R. and M.I.; visualization, M.I.; supervision, P.P.R. and B.-G.K.; project administration, B-G.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** No research funding has been received for this work.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.


**1. Introduction**

Since the onset of the Al-Aqsa Intifida in September of 2000, Israeli society has been witnessing continual terrorist attacks by Hamas and other terrorist organisations, including suicide bombings, drive-by shootings, knife and gun attacks, and missile attacks in urban settings launched from the Gaza strip. Hamas's ongoing threat against Israeli civilians has led to several military operations. The current study was conducted during Operation "Protective Edge", also known as the 2014 Gaza conflict. This operation was launched in the summer of 2014 in response to the substantial increase in Hamas's rocket attacks against Israeli communities, firing on an almost daily basis [1].

During Operation "Protective Edge", which lasted 50 days, more than 4500 rockets were launched towards Israel from Gaza. This operation and the period immediately preceding it represented an intense period of rocket and mortar fire against Israel's civilian population. Although the range of these rockets covered more than 70% of Israel's civilian population, those residing in communities near the Gaza Strip were most affected, having only 15 seconds to seek shelter. During this time, six civilians and 67 soldiers in Israel were killed, more than 1600 civilians were harmed, and an estimated 10,000 civilians evacuated their homes. In the Gaza Strip, approximately 2,125 Palestinians were killed [1].

Mental health professionals (MHPs) are among the first responders to address the needs of traumatized people following exposure to large-scale disasters, including terrorist attacks and wars. In the southern region of Israel, which has been subject to missile attacks from Gaza since 2001, MHPs encounter a double exposure to war-related trauma as community members and as professionals providing service to terror victims [2]. This situation in which MHPs are coping with the same traumatic event as their clients is referred to as "shared trauma," "shared tragedy," or "shared traumatic reality" [3–5]. Shared traumatic situations typically occur in communal disasters, such as natural disasters and war [6]. MHPs working in shared traumatic situations face multiple levels of vulnerability to traumatization, including direct, secondary, and vicarious traumatization [3].

The negative consequences of shared reality situations have been well documented. These consequences may include emotional distress during a traumatic event [6], as well as immediately after a traumatic event and up to a year later [5]. Cohen et al. [7], who interviewed therapists working with traumatized children following the shared traumatic reality of the Second Lebanon War, found high levels of anxiety, stress, and symptoms of posttraumatic stress disorder (PTSD) among the therapists. Similar findings emerged from Finklestein et al.'s [8] study of MHPs working in areas affected by repeated rocket attacks from the Gaza Strip, indicating that MHPs were at risk for both PTSD and vicarious trauma (VT) symptoms. Those who lived in the more affected area were at even greater risk for developing PTSD and VT symptoms. Additional studies have shown similar associations among level of exposure to terror attacks, PTSD symptoms, and emotional distress [6,9], providing further support for an incremental dose effect [6]. However, other studies have not found an association between exposure levels and emotional distress [10,11]. Increased levels of PTSD symptoms also have been reported among physicians and nurses exposed to a shared war-related reality in Israel [10,12,13] and in Gaza [14,15].

Work under shared reality conditions exposes MHPs to the blurring of boundaries between professional and personal lives [4,16,17], including boundaries between work and family loyalties [5,18]. Research also points to the blurring of boundaries between MHPs and their clients [5], manifested in their difficulty separating their personal experience from that of their clients [19].

Several studies on the effects of working in a shared war reality have reported a decrease in perceived professional competence among MHPs [5] and a sense of being deskilled [3]. However, other studies reveal a strong perception of professional competence [19] and high levels of professional confidence [9] among these MHPs.

Work in a shared traumatic reality also has been associated with positive consequences. These consequences may include a sense of growth, both personal and professional [4,10,16,19], and a sense of resilience [19]. Post-traumatic growth also has been reported among nurses working in a shared war–related reality in Israel [10] and in Gaza [15]. Positive consequences also can include heightened intimacy in the therapeutic relationship [4,20], a strong therapeutic alliance [19], a high level of work satisfaction, and a sense of agency and helpfulness [7].

The overall picture that emerges from studies of MHPs exposed to a shared traumatic reality stresses the importance of interventions designed to alleviate their emotional distress, particularly among those operating in areas highly exposed to armed conflicts and terror attacks. Interventions for MHPs in traumatic events mainly consist of group support [21,22], individual or group supervision [7], and debriefing sessions [7,9]. Research findings, however, have called into question the effectiveness of debriefing methods in alleviating symptoms of stress among MHPs and disaster workers [9,23]. One possible explanation for the ineffectiveness of debriefing methods is that MHPs view participation and sharing as integral to the organizational culture of mental health services, rather than as a unique intervention tailored to alleviate their war-related stress [9]. Another plausible explanation is that emotional turmoil and thoughts related to traumatic experiences do not lend themselves to easy verbalization [24]. The inadequacy of conventional verbal methods in the context of disasters points to the need to search for alternative methods of self-care for MHPs in shared war situations. To address this need, Huss, Sarid, and Cwikel [25] developed an art-based intervention model for stress reduction and self-care for social workers operating in a war zone during the Iron Cast Operation (2008). They based this intervention model on the use of a single drawing in a single group session. In the first stage of the intervention, social workers were asked to draw one image of their war experience as social workers. They were then instructed to identify the sources of their stress and their stress reactions within the artwork and to change their artwork by adding sources of coping and resilience. Allowing the social workers to change their artwork helped them to gain a sense of control over diffuse sources of anxiety. Huss and Sarid [26] have found that transformation of stressful visual images through drawing and in the imagination is linked to decreased levels of work-related stress among health care professionals. These findings demonstrate the efficacy of transforming a stressful image without extensive verbalization for stress reduction.

This article focuses on a CB-ART (cognitive behavioral and art-based) intervention for distress reduction that was developed based on these earlier findings. This intervention was implemented with MHPs who shared war-related experiences and distress with their clients during Operation "Protective Edge".