The Orthogonal Riesz Fractional Derivative

Abstract

The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials $C_n^{\left(\nu \right)}\left(x\right)$, where $\nu >-{\displaystyle \frac{1}{2}}$. Building on the work of Diekema and Koornwinder, the n-th derivative is obtained as the limit of an integral involving Gegenbauer polynomials as the kernel. When this limit is omitted, it results in the approximate Gegenbauer orthogonal derivative, which serves as an effective approximation of the n-th order derivative. Using this operator, we introduce a novel extension of the fractional Riesz derivative, denoted as ${}_x\mathcal{D}^{\alpha }$, providing an alternative framework for fractional calculus.

1. Introduction

In the work of Diekema and Koornwinder [1], a historically significant formula was revisited for what is known as the orthogonal derivative. This derivative, of order n, is computed as the limit of a specific integral, generalizing the traditional concept of the n-th order derivative. Remarkably, even without applying the limiting process, this formulation offers an effective approximation of the n-th order derivative. In a later development [2], the orthogonal derivative was used to extend the classical definitions of Riemann–Liouville and Weyl fractional derivatives [3], providing an efficient method to approximate these fractional derivatives. In particular, for the case involving Jacobi polynomials, the kernel of the associated integral transform that approximates the fractional derivative can be determined explicitly. Similarly, explicit results were derived for approximating fractional differences using Hahn polynomials. The objective of this paper is to extend the concept of the orthogonal derivative to obtain a new integral representation of the fractional Riesz derivative. In the one-dimensional case, the fractional Laplacian [4,5], often referred to as the Riesz fractional derivative, is widely used; see [6,7,8]. For clarity, we denote this operator as ${}_x\mathcal{D}^{\alpha }$ [9]. Interestingly, the fractional Laplacian and the Riesz fractional derivative share the same symbol, which can be expressed as follows [10]:

$${}_x\mathcal{D}^{\nu }={\mathcal{F}}^{-1}\left({\left({\xi }^2\right)}^{\nu /2}\mathcal{F}\right)=-{\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^{\nu /2}.$$

(1)

Building on the work of S. Bochner, who extended classical diffusion models to generalized diffusion equations for Lévy stable distributions, the fractional power of the second-order derivative can be written in an integral form valid for $\nu \in (0,2)$ [11]. This regularized form is expressed as

$$\begin{array}{cc}\hfill {}_x\mathcal{D}^{\nu }f\left(x\right)& =-{\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^{\nu /2}f\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{\Gamma (1+\nu )}{\pi }}sin\left({\displaystyle \frac{\nu \pi }{2}}\right){\int }_0^{\infty }{\displaystyle \frac{2f\left(x\right)-f(x+y)-f(x-y)}{y^{\nu +1}}}dy.\hfill \end{array}$$

(2)

It is crucial to highlight that the Riemann–Liouville and Weyl fractional derivatives, as defined by Diekema in [2], are fundamentally distinct from the Riesz derivative introduced in (2). These operators differ significantly in their symbolic representations, as outlined in [2] (Theorem 2.1). While the Riesz fractional derivative and the fractional Laplacian are known to coincide for functions $f\in \mathcal{S}\left(\mathbb{R}\right)$, this equivalence does not extend to functions defined on a proper subinterval of $\mathbb{R}$. For a comprehensive discussion on this discrepancy, see [12].

In this work, we focus on the orthogonal derivative associated with the Gegenbauer polynomials $C_n^{\left(\nu \right)}\left(x\right)$, where $\nu >-{\displaystyle \frac{1}{2}}$. Following the approach in [1], the n-th derivative is computed as the limit of a specific integral, with the Gegenbauer polynomials serving as the kernel. When the limiting process is omitted, this orthogonal derivative provides an approximation of the n-th order derivative, denoted by ${\mathcal{D}}_t^{\nu ,n}$, and referred to as the approximate Gegenbauer orthogonal derivative. This operator proves especially useful in cases where calculating the exact n-th order derivative is challenging or when numerical methods are required.

The key motivation for using the Gegenbauer orthogonal derivative, rather than Jacobi polynomials, lies in the advantageous symmetry properties of the Gegenbauer polynomials. Specifically, they satisfy the relation [13] (§9.8.1):

$$C_n^{\left(\nu \right)}(-x)={(-1)}^n{C}_n^{\left(\nu \right)}\left(x\right).$$

This symmetry enables an effective transfer function [2] for approximating the symbol of the Riesz derivative, ${(-{\xi }^2)}^{\nu /2}$ (for the definition of the transfer function, see [2]). This feature is not present when using Jacobi polynomials. Consequently, Gegenbauer polynomials provide a more efficient and accurate framework for these purposes.

This paper is structured as follows:

• Section 2 provides an overview of the notations and concepts related to the orthogonal derivative and Gegenbauer polynomials.
• Section 3 presents the main research outcomes, summarizing the key findings.
• Section 4 offers detailed proofs of the core results.

2. Preliminaries

We denote by $\mathcal{S}\left(\mathbb{R}\right)$ the Schwartz space of rapidly decreasing functions on $\mathbb{R}$. Let f be a function in $L^1\left(\mathbb{R}\right)$ (the space of integrable functions on $\mathbb{R}$). The Fourier transform $\mathcal{F}f$ of f is defined as

$$\left(\mathcal{F}f\right)\left(\xi \right)=\widehat{f}\left(\xi \right)={\int }_{\mathbb{R}}f\left(x\right)e^{-i\xi x}dx.$$

(3)

This transform has the following properties:

(i)

Inversion formula: The Fourier transform is a topological isomorphism from the Schwartz space $\mathcal{S}\left(\mathbb{R}\right)$ onto itself, and the function f can be recovered by the inversion formula:

$$f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}\widehat{f}\left(\xi \right)e^{i\xi x}d\xi ,f\in \mathcal{S}\left(\mathbb{R}\right).$$

(ii)

Convolution property: For $f,g\in \mathcal{S}\left(\mathbb{R}\right)$, the Fourier transform of the convolution of f and g satisfies

$$\widehat{f\ast g}=\widehat{f}\widehat{g},$$

where the convolution product $f\ast g$ is defined as

$$(f\ast g)\left(x\right)={\int }_{\mathbb{R}}f(x-y)g\left(y\right)dy.$$

In [1], the following theorem is established.

Theorem 1 

([1]). Let $p_n$ be an orthogonal polynomial of degree n with respect to the orthogonality measure μ. Suppose $x\in \mathbb{R}$, and let I be a closed interval such that, for some $ϵ>0$, $x+\delta \xi \in I$ for all $0\le \delta \le ϵ$ and $\xi \in supp\left(\mu \right)$. Assume that f is a continuous function on I and its derivatives of order $1,2,...,n$ exist at x. Additionally, if I is unbounded, assume that f has at most polynomial growth on I. Then,

$$f^{\left(n\right)}\left(x\right)=\underset{t\to 0^{+}}{lim}{\mathcal{D}}_t^{n}f\left(x\right),$$

(4)

where

$${\mathcal{D}}_t^{n}f\left(x\right)={\displaystyle \frac{k_{n}n!}{h_n}}{\displaystyle \frac{1}{t^n}}{\int }_{\mathbb{R}}f(x+ty)p_n\left(y\right)d\mu \left(y\right),$$

(5)

and the constants $h_n$ and $k_n$ are defined by

$$h_n:={\int }_{\mathbb{R}}{p}_n^2\left(x\right)d\mu \left(x\right),$$

and

$$p_n\left(x\right)=k_n{x}^n+\mathit{terms}\mathit{of}\mathit{degree}\mathit{less}\mathit{than}n,$$

where the integral converges absolutely.

In [1], the following version of Taylor’s Theorem is presented.

Proposition 1 

([1]). Let $x\in \mathbb{R}$ and let I be an interval containing x. Let f be a continuous function on I, such that its derivatives of order $1,2,...,n$ exist at x. Then, for small δ, we have the following expansion:

$$f(x+\delta )=\sum _{k=0}^n{\displaystyle \frac{f^{\left(k\right)}\left(x\right)}{k!}}{\delta }^k+{\delta }^n{F}_{x,n}\left(\delta \right),$$

where $F_{x,n}$ is continuous on $I-x$ and satisfies $F_{x,n}\left(0\right)=0$. Moreover, $F_{x,n}$ is bounded on $I-x$ if f is bounded on I. Finally, if I is unbounded and f exhibits polynomial growth on I, then $F_{x,n}$ is also polynomial growth on $I-x$.

Next, we can apply Formula (4) to the Gegenbauer polynomials $C_n^{\left(\nu \right)}\left(x\right)$. Before doing so, let us first recall the definition and some essential properties of the Gegenbauer polynomials. The Gegenbauer (or ultraspherical) polynomials are defined as

$$C_n^{\left(\nu \right)}\left(x\right)={\displaystyle \frac{{\left(2\nu \right)}_n}{n!}}{}_{2}F_1\left(\begin{array}{c}-n,n+2\nu \\ \nu +{\displaystyle \frac{1}{2}}\end{array};{\displaystyle \frac{1-x}{2}}\right),$$

which can also be expressed as

$$C_n^{\left(\nu \right)}\left(x\right)={\displaystyle \frac{{\left(2\nu \right)}_n}{n!}}\sum _{k=0}^n{\displaystyle \frac{{(-n)}_k{(n+2\nu )}_k}{{(\nu +\frac{1}{2})}_{k}k!}}{\left({\displaystyle \frac{1-x}{2}}\right)}^k,$$

(6)

where ${}_{2}F_1$ represents the hypergeometric function, and ${\left(a\right)}_n$ is the Pochhammer symbol (falling factorial) of the complex number a, defined as

$${\left(a\right)}_n=a(a+1)(a+2)\cdots (a+n-1)={\displaystyle \frac{\Gamma (a+n)}{\Gamma \left(a\right)}},$$

for $n\ge 0$, with ${\left(a\right)}_0=1$.

The Gegenbauer polynomials are critical in various areas of mathematical physics and approximation theory due to their orthogonality and recurrence relations. Their connection with hypergeometric functions makes them especially useful in solving differential equations and expanding functions in series.

For $\nu >-{\displaystyle \frac{1}{2}}$ and $\nu \ne 0$, the Gegenbauer polynomials $C_n^{\left(\nu \right)}\left(x\right)$ satisfy the following orthogonality condition:

$${\int }_{-1}^1{(1-x^2)}^{\nu -{\displaystyle \frac{1}{2}}}{C}_m^{\left(\nu \right)}\left(x\right)C_n^{\left(\nu \right)}\left(x\right)dx=h_n{\delta }_{mn},$$

where $h_n$ is given by

$$h_n=\pi {\displaystyle \frac{\Gamma (n+2\nu )2^{1-2\nu }}{{\left[\Gamma \left(\nu \right)\right]}^2(n+\nu )n!}}.$$

(7)

From (6), it follows that

$$C_n^{\left(\nu \right)}\left(x\right)={\displaystyle \frac{2^n{\left(\nu \right)}_n}{n!}}{x}^n+\mathrm{terms}\mathrm{of}\mathrm{degree}\mathrm{less}\mathrm{than}n.$$

(8)

The corresponding transform associated with the Gegenbauer polynomials, denoted by ${\mathcal{D}}_t^{\nu ,n}f\left(x\right)$, is referred to as the approximate Gegenbauer orthogonal derivative. After some straightforward computations, we can obtain the following expression for $\nu >-{\displaystyle \frac{1}{2}}$ and $n\in {\mathbb{N}}_0$:

$${\mathcal{D}}_t^{\nu ,n}f\left(x\right)={\displaystyle \frac{{\gamma }_{\nu ,n}}{t^n}}{\int }_{-1}^{1}f(x+ty)C_n^{\left(\nu \right)}\left(y\right){\left(1-y^2\right)}^{\nu -{\displaystyle \frac{1}{2}}}dy,$$

(9)

where ${\gamma }_{\nu ,n}$ is defined as

$${\gamma }_{\nu ,n}={\displaystyle \frac{k_{n}n!}{h_n}}={\displaystyle \frac{2^{2\nu +n-1}n!\Gamma \left(\nu \right)\Gamma (n+\nu +1)}{\pi \Gamma (n+2\nu )}}.$$

(10)

Using the operator ${\mathcal{D}}_t^{\nu ,n}$, we define the local Diekema–Koornwinder orthogonal derivative as

$$f^{\left[n\right]}\left(x\right)=\underset{t\to 0^{+}}{lim}{\mathcal{D}}_t^{\nu ,n}f\left(x\right),$$

(11)

whenever this limit exists. The notation $f^{\left[n\right]}\left(x\right)$ is used to differentiate this derivative from the standard derivative $f^{\left(n\right)}\left(x\right)$. For a function $f\in C^{\left(n\right)}\left([-1,1]\right)$, from [1] (Theorem 3.2), we have

$$f^{\left(n\right)}\left(x\right)=\underset{t\to 0^{+}}{lim}{\mathcal{D}}_t^{\nu ,n}f\left(x\right).$$

(12)

In this case, the local Diekema–Koornwinder orthogonal derivative $f^{\left[n\right]}\left(x\right)$ coincides with the n-th derivative $f^{\left(n\right)}\left(x\right)$. However, unlike the classical derivative, this operator can be applied to functions that are not necessarily smooth.

Alternatively, the operator ${\mathcal{D}}_t^{\nu ,n}f\left(x\right)$ can be expressed as a convolution:

$${\mathcal{D}}_t^{\nu ,n}f\left(x\right)=(w_t^{\nu ,n}\ast f)\left(x\right),$$

(13)

where

$$w_t^{\nu ,n}\left(x\right)=t^{-n-1}{w}^{\nu ,n}\left(-{\displaystyle \frac{x}{t}}\right),t>0,x\in \mathbb{R},$$

and the function $w^{\nu ,n}\left(y\right)$ is given by

$$w^{\nu ,n}\left(x\right)={\gamma }_{\nu ,n}\left\{\begin{array}{cc}{(1-x^2)}^{\nu -{\displaystyle \frac{1}{2}}}{C}_n^{\left(\nu \right)}\left(x\right),\hfill & \mathrm{if}\left|x\right|<1,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(14)

We also require the Gegenbauer generalization of Poisson’s integral formula [14] (§3.32), which is given by

$$J_{\nu +n}\left(x\right)={\displaystyle \frac{{(-i)}^{n}n!{\left(\frac{x}{2}\right)}^{\nu }}{\Gamma (\nu +1/2)\Gamma (1/2)\Gamma (2\nu +n)}}{\int }_{-1}^1{e}^{ixt}{\left(1-t^2\right)}^{\nu -1/2}{C}_n^{\left(\nu \right)}\left(t\right)dt,$$

(15)

where $J_{\nu }\left(x\right)$ is the Bessel function of the first kind [14].

Proposition 2. 

Let $\nu >-n-{\displaystyle \frac{1}{2}}$, where $n\in {\mathbb{N}}_0$, and let $f\in \mathcal{S}\left(\mathbb{R}\right)$. Then, for $t>0$, the Fourier transform of ${\mathcal{D}}_t^{\nu ,n}f$ is given by

$$\widehat{{\mathcal{D}}_t^{\nu ,n}f}\left(\xi \right)={\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)\widehat{f}\left(\xi \right),$$

where ${\mathcal{J}}_{\nu +n}\left(t\xi \right)$ denotes the normalized Bessel function of order $\nu +n$, defined as

$${\mathcal{J}}_{\nu }\left(x\right):=\Gamma (\nu +1){\left({\displaystyle \frac{2}{x}}\right)}^{\nu }{J}_{\nu }\left(x\right),\nu >-1.$$

(16)

Proof. 

Using Gegenbauer’s generalization of Poisson’s integral formula (15), we have

$${\left(it\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)={\displaystyle \frac{2^{n}n!\Gamma \left(2\nu \right)\Gamma (n+\nu +1)}{\sqrt{\pi }\Gamma \left(\nu +\frac{1}{2}\right)\Gamma (2\nu +n)}}{\int }_{-1}^1{e}^{it\xi u}{\left(1-u^2\right)}^{\nu -{\displaystyle \frac{1}{2}}}{C}_n^{\left(\nu \right)}\left(u\right)du.$$

(17)

This expression shows that ${\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)$ is equal to the Fourier transform of the function $w_t^{\nu ,n}\left(x\right)$. More specifically, it follows that

$${\left(it\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)=\widehat{w_t^{\nu ,n}}\left(\xi \right).$$

Therefore, applying this result, we conclude that

$$\widehat{{\mathcal{D}}_t^{\nu ,n}f}\left(\xi \right)=\left(\widehat{w_t^{\nu ,n}\ast f}\right)\left(\xi \right)=\widehat{w_t^{\nu ,n}}\left(\xi \right)\widehat{f}\left(\xi \right)={\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)\widehat{f}\left(\xi \right),$$

which proves the proposition. □

Remark 1. 

We can extend the definition of the approximate Gegenbauer orthogonal derivative to a general complex parameter ν. In this context, the operator can be interpreted as a Fourier multiplier, expressed as follows:

$$\widehat{{\mathcal{D}}_t^{\nu ,n}f}\left(\xi \right)={\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)\widehat{f}\left(\xi \right).$$

(18)

In the terminology of filter theory, the function $H_{n,\nu }\left(\xi \right)={\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)$ is called the transfer function [2]. Since the normalized Bessel function ${\mathcal{J}}_{\nu }\left(\xi \right)$ is even, the transfer function $H_{0,\nu }\left(\xi \right)$ is also even. However, this property does not hold when using Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$. For the evaluation of the transform for the approximate Jacobi orthogonal derivative, see formula (29) in [2].

3. Statement of the Main Results

In this section, we present the main results of the paper.

Theorem 2. 

For $\nu \ge 0$, $0<s<1$, and for $f\in \mathcal{S}\left(\mathbb{R}\right)$, the following equality holds:

$$\begin{array}{cc}\hfill {\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^s\left({\displaystyle \frac{d^{n}f}{d{x}^n}}\right)\left(x\right)& ={\lambda }_n(s,\nu ){\int }_0^{\infty }{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^{1+2s}}}dt,\hfill \end{array}$$

(19)

where the normalization constant ${\lambda }_n(s,\alpha )$ is given by

$${\lambda }_n(s,\nu )={\displaystyle \frac{2^{1+2s}\Gamma \left(\nu +n+s+1\right)}{\Gamma \left(\nu +n+1\right)|\Gamma (-s)|}}.$$

(20)

In particular, for $\nu =-{\displaystyle \frac{1}{2}}$ and $n=0$, the approximate Gegenbauer orthogonal derivative becomes the average operator,

$${\mathcal{D}}_t^{-1/2,0}f\left(x\right)={\mathcal{M}}_{t}f\left(x\right),$$

where

$${\mathcal{M}}_{t}f\left(x\right)={\displaystyle \frac{f(x+t)+f(x-t)}{2}}.$$

As a direct consequence of Theorem 2, in the case where $\nu =-{\displaystyle \frac{1}{2}}$ and $n=0$, we obtain a formula for the well-known fractional Riesz derivative:

$$\begin{array}{cc}\hfill {\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^{s}f\left(x\right)& ={\displaystyle \frac{2\Gamma (1+2s)}{\pi }}sin\left(\pi s\right){\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-{\mathcal{M}}_{t}f\left(x\right)}{t^{1+2s}}}dt\hfill \\ \hfill & ={\displaystyle \frac{\Gamma (1+2s)}{\pi }}sin\left(\pi s\right){\int }_0^{\infty }{\displaystyle \frac{2f\left(x\right)-f(x+t)-f(x-t)}{t^{1+2s}}}dt.\hfill \end{array}$$

For $\nu ={\displaystyle \frac{1}{2}}$ and $n=0$, the approximate Gegenbauer orthogonal derivative reduces to the solid average operator:

$${\mathcal{D}}_t^{1/2,0}f\left(x\right)={\mathcal{A}}_{t}f\left(x\right),$$

where

$${\mathcal{A}}_{t}f\left(x\right)={\displaystyle \frac{1}{2t}}{\int }_{x-t}^{x+t}f\left(u\right)du.$$

As a consequence of Theorem 2, we can obtain the following corollary:

Corollary 1. 

Let $0<s<1$. For $f\in \mathcal{S}\left(\mathbb{R}\right)$, the fractional Riesz derivative of f can be expressed in terms of the solid mean-value operator, as

$$\begin{array}{cc}\hfill {\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^{s}f\left(x\right)& ={\displaystyle \frac{2^{2s+2}\Gamma \left(s+\frac{3}{2}\right)}{\sqrt{\pi }|\Gamma (-s)|}}{\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-\frac{1}{2t}{\int }_{x-t}^{x+t}f\left(u\right)du}{t^{1+2s}}}dt.\hfill \end{array}$$

(21)

For $\nu ={\displaystyle \frac{1}{2}}$ and $n=1$, the approximate Gegenbauer orthogonal derivative reduces to the Lanczos operator:

$${\mathcal{D}}_t^{1/2,1}f\left(x\right)={\mathcal{L}}_{t}f\left(x\right),$$

where

$${\mathcal{L}}_{t}f\left(x\right)={\displaystyle \frac{3}{2t}}{\int }_{-1}^{1}f(x+tu)udu.$$

As a result, we can derive the following corollary:

Corollary 2. 

Let $0<s<1$. For $f\in \mathcal{S}\left(\mathbb{R}\right)$, the fractional Riesz derivative of f can be expressed in terms of the Lanczos operator as

$$\begin{array}{cc}\hfill {\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^s{f}^{\prime }\left(x\right)& ={\displaystyle \frac{2^{2s+3}\Gamma \left(s+\frac{5}{2}\right)}{3\sqrt{\pi }|\Gamma (-s)|}}{\int }_0^{\infty }{\displaystyle \frac{f^{\prime }\left(x\right)-\frac{3}{2t}{\int }_{-1}^{1}f(x+tu)udu}{t^{1+2s}}}dt.\hfill \end{array}$$

(22)

4. Proof of the Main Results

Before presenting the proof of the main results, we first need to establish some additional lemmas to prepare for the proof.

Lemma 1 

([1]). Let I be an interval, and let $x\in I$. Suppose that $f\in C^n\left(I\right)$. Then,

$$f(x+h)=\sum _{k=0}^n{\displaystyle \frac{f^{\left(k\right)}\left(x\right)}{k!}}{h}^k+h^n{F}_{x,n}\left(h\right),$$

where

$$F_{x,n}\left(h\right)={\displaystyle \frac{1}{(n-1)!}}{\int }_0^1\left(f^{\left(n\right)}(x+th)-f^{\left(n\right)}\left(x\right)\right){(1-t)}^{n-1}dt.$$

(23)

Furthermore, if $f^{\left(n\right)}$ is of polynomial growth on I, then $F_{x,n}\left(h\right)\to 0$ as $h\to 0$ uniformly for x in compact subsets of I.

Lemma 2. 

For $\nu >-{\displaystyle \frac{1}{2}}$ and $\nu \ne 0$, for all $n\in \mathbb{N}$, the following hold:

(i) 

$${\int }_{-1}^1{y}^n{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy={\displaystyle \frac{\pi \Gamma (n+2\nu )}{2^{n+2\nu -1}{\Gamma }^2\left(\nu \right){\left(\nu \right)}_{n+1}}}.$$

(ii) 

$${\int }_{-1}^1{y}^{n+1}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy=0.$$

(iii) 

$${\int }_{-1}^1{y}^{n+2}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy={\displaystyle \frac{\sqrt{\pi }(n+2)(n+1){\left(2\nu \right)}_n\Gamma (\nu +\frac{1}{2})}{2^{n+1}\Gamma (\nu +n+2)}}.$$

Proof. 

To prove part (i), we begin by utilizing the orthogonality relation for Gegenbauer polynomials along with the normalization constant provided in Equation (7). Since the Gegenbauer polynomial $C_n^{\left(\nu \right)}\left(x\right)$ has the expansion

$$C_n^{\left(\nu \right)}\left(x\right)={\displaystyle \frac{2^n{\left(\nu \right)}_n}{n!}}{x}^n+\mathrm{terms}\mathrm{of}\mathrm{degree}\mathrm{less}\mathrm{than}n,$$

(24)

we know that the orthogonality relation implies

$${\int }_{-1}^1{y}^k{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy=0,\mathrm{for}k=0,...,n-1.$$

This allows us to focus on the term corresponding to $y^n$, leading to

$$\begin{array}{cc}\hfill {\int }_{-1}^1{y}^n{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy& ={\displaystyle \frac{n!}{2^n{\left(\nu \right)}_n}}{\int }_{-1}^1{\left(C_n^{\left(\nu \right)}\left(y\right)\right)}^2{(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy\hfill \\ \hfill & ={\displaystyle \frac{\pi \Gamma (n+2\nu )}{2^{n+2\nu -1}{\Gamma }^2\left(\nu \right){\left(\nu \right)}_{n+1}}}.\hfill \end{array}$$

For part (ii), using the property $C_n^{\left(\nu \right)}(-y)={(-1)}^n{C}_n^{\left(\nu \right)}\left(y\right)$, we have

$${\int }_{-1}^1{y}^{n+1}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy=0.$$

For part (iii), we begin by using the Rodrigues-type formula for Gegenbauer polynomials [13] (formula 9.8.27):

$${(1-x^2)}^{\nu -{\displaystyle \frac{1}{2}}}{C}_n^{\left(\nu \right)}\left(x\right)={\displaystyle \frac{{(-1)}^n{\left(2\nu \right)}_n}{2^{n}n!{(\nu +\frac{1}{2})}_n}}{\displaystyle \frac{d^n}{d{x}^n}}\left[{(1-x^2)}^{\nu +n-{\displaystyle \frac{1}{2}}}\right].$$

We also have

$${\int }_{-1}^1{y}^{n+2}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy={\displaystyle \frac{{(-1)}^n{\left(2\nu \right)}_n}{2^{n}n!{(\nu +\frac{1}{2})}_n}}{\int }_{-1}^1{y}^{n+2}{\displaystyle \frac{d^n}{d{y}^n}}\left[{(1-y^2)}^{\nu +n-{\displaystyle \frac{1}{2}}}\right]dy.$$

At this point, we apply n-fold integration by parts. Each time we integrate by parts, we reduce the power of $y^{n+2}$ in the integrand, transferring derivatives to the function ${(1-y^2)}^{\nu +n-{\displaystyle \frac{1}{2}}}$. After completing this process, all derivatives will have been applied, and we will be left with the following expression:

$${\displaystyle \frac{(n+2)(n+1){\left(2\nu \right)}_n}{2^n{(\nu +\frac{1}{2})}_n}}{\int }_{-1}^1{y}^2{(1-y^2)}^{\nu +n-{\displaystyle \frac{1}{2}}}dy.$$

To compute this integral, we make the substitution $u=y^2$, so that $du=2ydy$, transforming the integral as follows:

$${\int }_{-1}^1{y}^2{(1-y^2)}^{\nu +n-{\displaystyle \frac{1}{2}}}dy={\int }_0^1{u}^{{\displaystyle \frac{1}{2}}}{(1-u)}^{\nu +n-{\displaystyle \frac{1}{2}}}du.$$

This integral can be expressed in terms of the Beta function $B(x,y)$, which is given by

$$B(x,y)={\int }_0^1{u}^{x-1}{(1-u)}^{y-1}du={\displaystyle \frac{\Gamma \left(x\right)\Gamma \left(y\right)}{\Gamma (x+y)}}.$$

In our case, we have $x={\displaystyle \frac{3}{2}}$ and $y=\nu +n+{\displaystyle \frac{1}{2}}$, so the integral becomes

$${\int }_0^1{u}^{{\displaystyle \frac{1}{2}}}{(1-u)}^{\nu +n-{\displaystyle \frac{1}{2}}}du=B\left({\displaystyle \frac{3}{2}},\nu +n+{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{\Gamma \left(\frac{3}{2}\right)\Gamma \left(\nu +n+\frac{1}{2}\right)}{\Gamma (\nu +n+2)}}.$$

Substituting this result back into the original expression, we obtain

$${\int }_{-1}^1{y}^{n+2}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy={\displaystyle \frac{\sqrt{\pi }(n+2)(n+1){\left(2\nu \right)}_n\Gamma (\nu +\frac{1}{2})}{2^{n+1}\Gamma (\nu +n+2)}}.$$

Thus, we have completed the proof for part (iii). □

Lemma 3. 

Let $f\in C^{n+2}\left([-1,1]\right)$. Then,

$$f^{(n+2)}\left(x\right)=-2(n+\nu +1)\underset{t\to 0^{+}}{lim}{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^2}}.$$

Proof. 

Since $f\in C^{n+2}\left([-1,1]\right)$, from Proposition 1, we have

$$f(x+ty)=\sum _{k=0}^{n+2}{\displaystyle \frac{f^{\left(k\right)}\left(x\right)}{k!}}{\left(ty\right)}^k+{\left(ty\right)}^{n+2}{F}_{x,n+2}\left(ty\right).$$

(25)

Inserting Equation (25) into ${\mathcal{D}}_t^{\alpha ,n}f\left(x\right)$, we obtain

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_t^{\nu ,n}f\right)\left(x\right)& ={\displaystyle \frac{{\gamma }_{\nu ,n}}{t^n}}{\int }_{-1}^{1}f(x+ty)C_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy\hfill \\ \hfill & ={\displaystyle \frac{{\gamma }_{\nu ,n}}{t^n}}\sum _{k=0}^{n+2}{t}^k{\displaystyle \frac{f^{\left(k\right)}\left(x\right)}{k!}}{\int }_{-1}^1{y}^k{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_{\nu ,n}}{t^n}}{\int }_{-1}^1{F}_{x,n+2}\left(ty\right)C_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy.\hfill \end{array}$$

By using the orthogonality relations for the Gegenbauer polynomials, we obtain

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_t^{\nu ,n}f\right)\left(x\right)& ={\gamma }_{\nu ,n}{\displaystyle \frac{f^{\left(n\right)}\left(x\right)}{n!}}{\int }_{-1}^1{y}^n{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy\hfill \\ \hfill & +{\gamma }_{\nu ,n}t{\displaystyle \frac{f^{(n+1)}\left(x\right)}{(n+1)!}}{\int }_{-1}^1{y}^{n+1}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy\hfill \\ \hfill & +{\gamma }_{\nu ,n}{t}^2{\displaystyle \frac{f^{(n+2)}\left(x\right)}{(n+2)!}}{\int }_{-1}^1{y}^{n+2}{C}_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy\hfill \\ \hfill & +{\gamma }_{\nu ,n}{t}^2{\int }_{-1}^1{y}^{n+2}{F}_{x,n+2}\left(ty\right)C_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy.\hfill \end{array}$$

Therefore, using Lemma 2, we obtain

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_t^{\nu ,n}f\right)\left(x\right)& =f^{\left(n\right)}\left(x\right)+{\displaystyle \frac{t^2}{2(n+\nu +1)}}{f}^{(n+2)}\left(x\right)\hfill \\ & +{\gamma }_{\nu ,n}{t}^2{\int }_{-1}^1{y}^{n+2}{F}_{x,n+2}\left(ty\right)C_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy.\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill f^{(n+2)}\left(x\right)& =-2(n+\nu +1){\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^2}}\hfill \\ & -2(n+\nu +1){\gamma }_{\nu ,n}{\int }_{-1}^1{y}^{n+2}{F}_{x,n+2}\left(ty\right)C_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy.\hfill \end{array}$$

Finally, from Lemma 1, we know that

$$\underset{t\to 0^{+}}{lim}{\int }_{-1}^1{y}^{n+2}{F}_{x,n+2}\left(ty\right)C_n^{\left(\nu \right)}\left(y\right){(1-y^2)}^{\nu -{\displaystyle \frac{1}{2}}}dy=0.$$

This completes the proof of the lemma. □

Lemma 4. 

Let $f\in \mathcal{S}\left(\mathbb{R}\right)$. The following inequality holds:

$${\parallel f^{\left(n\right)}-{\mathcal{D}}_t^{\nu ,n}f\parallel }_{\infty }\le {\pi }^{-1}{\parallel {(·)}^n\widehat{f}\parallel }_{L^1\left(\mathbb{R}\right)}.$$

Proof. 

According to Proposition 2, the approximate Gegenbauer orthogonal derivative ${\mathcal{D}}_t^{\nu ,n}f\left(x\right)$ can be expressed in terms of the Fourier transform as

$$\begin{array}{cc}\hfill {\mathcal{D}}_t^{\nu ,n}f\left(x\right)& =\left({\mathcal{F}}^{-1}\left[{\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)\mathcal{F}f\left(\xi \right)\right]\right)\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)\mathcal{F}f\left(\xi \right)e^{ix\xi }d\xi .\hfill \end{array}$$

(26)

According to the Fourier inversion formula, we also have

$$f^{\left(n\right)}\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n\mathcal{F}f\left(\xi \right)e^{ix\xi }d\xi .$$

(27)

Subtracting (27) from (26), we can obtain

$$\begin{array}{c}\hfill f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n\left(1-{\mathcal{J}}_{\nu +n}\left(t\xi \right)\right)\mathcal{F}f\left(\xi \right)e^{ix\xi }d\xi .\end{array}$$

(28)

Using the well-known inequality for the normalized Bessel function ${\mathcal{J}}_{\nu +n}$, we have [14]

$$\left|{\mathcal{J}}_{\nu +n}\right(x\left)\right|\le 1.$$

Thus, we can write

$$\begin{array}{cc}\hfill |f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)|& \le {\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left|\xi \right|}^n\left(1+|{\mathcal{J}}_{\nu +n}\left(t\xi \right)|\right)\left|\mathcal{F}f\left(\xi \right)\right|d\xi \hfill \\ \hfill & \le 2·{\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left|\xi \right|}^n\left|\mathcal{F}f\left(\xi \right)\right|d\xi \hfill \\ \hfill & \le {\pi }^{-1}{\parallel {(·)}^n\widehat{f}\parallel }_{L^1\left(\mathbb{R}\right)}.\hfill \end{array}$$

(29)

□

Lemma 5 

([15]). Let $x\in \mathbb{R},$$\nu \ge -1/2$ and $0<s<2$; we have

$${\left|x\right|}^s={\displaystyle \frac{2^{s+1}\Gamma (\nu +\frac{s}{2}+1)}{\Gamma (\nu +1)|\Gamma (-\frac{s}{2}\left)\right|}}{\int }_0^{\infty }\left(1-{\mathcal{J}}_{\nu }\left(tx\right)\right){\displaystyle \frac{dt}{t^{s+1}}}.$$

Now, we can proceed to the proof of Theorem 2.

Proof of Theorem 2. 

Let $f\in \mathcal{S}\left(\mathbb{R}\right)$ be a Schwartz function. According to Lemma 3, we have the following relation:

$$f^{(n+2)}\left(x\right)=-2(n+\nu +1)\underset{t\to 0}{lim}{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^2}}.$$

This shows that the integrand in the right-hand side of (19) behaves as

$${\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^{1+2s}}}=O\left(t^{1-2s}\right)\mathrm{as}t\to 0.$$

For $s\in (0,1)$, this implies convergence of the integral near $t=0$.

Next, from Lemma 4, we know that

$${\int }_1^{\infty }{\displaystyle \frac{|f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)|}{t^{1+2s}}}dt\le {\pi }^{-1}{\parallel {(·)}^n\widehat{f}\parallel }_{L^1\left(\mathbb{R}\right)}{\int }_1^{\infty }{\displaystyle \frac{1}{t^{1+2s}}}dt.$$

Now, computing the integral ${\int }_1^{\infty }{\displaystyle \frac{1}{t^{1+2s}}}dt$, we can obtain

$${\int }_1^{\infty }{\displaystyle \frac{1}{t^{1+2s}}}dt={\displaystyle \frac{1}{2s}}.$$

Thus, the integral converges near $t=\infty$, and we have

$${\int }_1^{\infty }{\displaystyle \frac{|f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)|}{t^{1+2s}}}dt\le {\displaystyle \frac{{\parallel {(·)}^n\widehat{f}\parallel }_{L^1\left(\mathbb{R}\right)}}{2\pi s}}.$$

The approximate Gegenbauer orthogonal derivative ${\mathcal{D}}_t^{\nu ,n}f\left(x\right)$ can be written in terms of the Fourier transform, as

$${\mathcal{D}}_t^{\nu ,n}f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n{\mathcal{J}}_{\nu +n}\left(t\xi \right)\widehat{f}\left(\xi \right)e^{ix\xi }d\xi .$$

According to the Fourier inversion formula,

$$f^{\left(n\right)}\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n\widehat{f}\left(\xi \right)e^{ix\xi }d\xi .$$

Subtracting the two expressions gives

$$f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n\left(1-{\mathcal{J}}_{\nu +n}\left(t\xi \right)\right)\widehat{f}\left(\xi \right)e^{ix\xi }d\xi .$$

Dividing by $t^{1+2s}$ and integrating with respect to t, we obtain

$${\int }_0^{\infty }{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^{1+2s}}}dt={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n\widehat{f}\left(\xi \right)e^{ix\xi }\left({\int }_0^{\infty }{\displaystyle \frac{1-{\mathcal{J}}_{\nu +n}\left(t\xi \right)}{t^{1+2s}}}dt\right)d\xi .$$

Using Lemma 5, we know that

$${\left|\xi \right|}^{2s}={\lambda }_n(s,\nu ){\int }_0^{\infty }{\displaystyle \frac{1-{\mathcal{J}}_{\nu +n}\left(t\xi \right)}{t^{1+2s}}}dt,0<s<1,$$

where

$${\lambda }_n(s,\nu )={\displaystyle \frac{2^{1+2s}\Gamma \left(\nu +n+s+1\right)}{\Gamma \left(\nu +n+1\right)|\Gamma (-s)|}}.$$

Substituting this into the equation, we obtain

$${\int }_0^{\infty }{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^{1+2s}}}dt={\displaystyle \frac{1}{{\lambda }_n(s,\nu )2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n{\left|\xi \right|}^{2s}\widehat{f}\left(\xi \right)e^{ix\xi }d\xi .$$

Multiplying both sides by ${\lambda }_n(s,\nu )$, we have

$${\lambda }_n(s,\nu ){\int }_0^{\infty }{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^{1+2s}}}dt={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left(i\xi \right)}^n{\left|\xi \right|}^{2s}\widehat{f}\left(\xi \right)e^{ix\xi }d\xi ,$$

which is precisely the Fourier representation of the fractional Laplacian applied to $f^{\left(n\right)}\left(x\right)$. Therefore, we obtain

$${\lambda }_n(s,\nu ){\int }_0^{\infty }{\displaystyle \frac{f^{\left(n\right)}\left(x\right)-{\mathcal{D}}_t^{\nu ,n}f\left(x\right)}{t^{1+2s}}}dt={\left(-{\displaystyle \frac{d^2}{d{x}^2}}\right)}^s\left({\displaystyle \frac{d^{n}f}{d{x}^n}}\right)\left(x\right),$$

completing the proof. □

5. Conclusions

In this paper, we have extended the concept of the orthogonal derivative, originally developed by Diekema and Koornwinder, to derive new integral representations for the fractional Riesz derivative. Building on prior work that bridges classical and fractional derivatives, we have demonstrated that Gegenbauer polynomials offer a robust and efficient framework for approximating fractional derivatives. Their symmetry properties, particularly in contrast to Jacobi polynomials, make them especially well-suited for the Riesz derivative.