Novel Results on Legendre Polynomials in the Sense of a Generalized Fractional Derivative

Abstract

In this article, new results are investigated in the context of the recently introduced Abu-Shady–Kaabar fractional derivative. First, we solve the generalized Legendre fractional differential equation. As in the classical case, the generalized Legendre polynomials constitute notable solutions to the aforementioned fractional differential equation. In the sense of the fractional derivative of Abu-Shady–Kaabar, we establish important properties of the generalized Legendre polynomials such as Rodrigues formula and recurrence relations. Special attention is also devoted to another very important property of Legendre polynomials and their orthogonal character. Finally, the representation of a function $f\in L_{\alpha }^2\left(\left[-1,1\right]\right)$ in a series of generalized Legendre polynomials is addressed.

1. Introduction

In theory, fractional-order calculus is considered as a naturally extended form of classical derivatives. A lot of research studies have been conducted on this topic due to its powerful applicability in modelling phenomena in natural sciences and engineering such as the Klein–Fock–Gordon equation [1], Hepatitis B model [2], and other partial differential equations [3,4,5,6]. Fixed point techniques have been recently studied in detail [7] via fractional operators (see also [8]). Many of the classical nonlocal fractional-order definitions such as Liouville–Caputo and Riemann–Liouville (see [9,10] for more details) try to satisfy the essential properties of the classical derivative, but they cannot satisfy them, other than the property that is inherent in these definitions which is the linearity property.

For the locally defined derivatives, certain quotients of increments are employed in proposing such definitions. One notable example of such derivatives is the conformable derivative, proposed by Khalil et al. [11], to obtain analytical solutions for differential equations in the sense of conformable calculus. The physical and geometric interpretations of the conformable derivative are mentioned in [12,13], respectively. However, this definition suffers from some drawbacks which have been highlighted in [14] in comparison with the obtained results from the Liouville–Caputo definition.

A recently generalized definition of fractional derivative (GDFD), proposed by Abu-Shady and Kaabar [15], is successfully employed in various fractional differential equations to simply obtain analytical solutions which are in agreement with the results obtained using the Liouville–Caputo and Riemann–Liouville formulations. In addition, GDFD overcomes all challenges and disadvantages associated with a conformable definition and some other fractional derivatives. In [16,17], the authors have established new results that complete the GDFD theory. Among these important results established in the GDFD, it is worth noting that the chain rule, the derivation of the inverse function, the Barrow´s rule or some properties of the modules of the generalized $\alpha$-integral, the development of the theory of fractional power series, and the search for solutions of the generalized fractional Chebyshev differential equation of first kind using the fractional power series method, among other results, are studied. In addition, GDFD has been applied recently in studying the diatomic molecules in the Deng–Fan model [18] and heavy tetraquark masses spectra via the extended Nikiforov–Uvaro technique [19].

It is a well-known fact that the Legendre polynomials represent one of the most relevant special functions in mathematical physics. The classic Legendre polynomials have important applications in mathematical physics, and these applications depend on their properties [20]. Among these important applications, we can highlight the calculation of the gravitational (or electrostatic) potential generated by a particle (or charge) located at a certain point of space, the calculation and analysis of the tidal phenomenon, or the stationary temperature in a sphere [21,22]. Some recent studies have utilized Legendre functions in various applications such as the coupled Lane–Emden equations [23] and logistic equation [24] in the sense of the Liouville–Caputo fractional derivative. The aim of this article is to investigate the properties of Legendre polynomials in the context of GFDF theory. Thus, we will study the following topics:

(i)

Using the fractional power series method, we solve the following generalized fractional Legendre differential equation:

$${\left(1-t^{2\alpha }\right)}^{\left(2\right)}{D}^{GDFD}y\left(t\right)-\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t^{\alpha }}^{\left(1\right)}{D}^{GDFD}y\left(t\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}p\left(p+1\right)y\left(t\right)=0,$$

where $\alpha \in (0,1]$, $t>0,$ and $p$ is a real constant. Gamma function is considered as one of the most essential special functions in the theory of fraction calculus where it is defined as $\mathsf{\Gamma }\left(\varphi \right)·{{\displaystyle \int }}_0^{\infty }{e}^{-t}{t}^{\varphi -1}dt$ for all $Re\left\{\varphi \right\}>0$, and some of its properties are $\mathsf{\Gamma }\left(\varphi +1\right)=\varphi \mathsf{\Gamma }\left(\varphi \right)$ and $\mathsf{\Gamma }\left(w+1\right)=w!$ such that $w\in \mathbb{N}$. In the case that the parameter $p$ is a non-negative integer, there exists a polynomial solution for the generalized Legendre equation. These polynomial solutions, when certain additional conditions are imposed on them, form a set of polynomials, which we will call generalized Legendre polynomials.

(ii)

The Rodrigues formula is established in the context of the GDFD, which constitutes a generating formula of the generalized Legendre polynomials. In addition, through the aforementioned formula, we guarantee the existence of these polynomials, at the same time that it provides us with an argument to prove their uniqueness.

(iii)

As in the classical case, some useful recurrence relations that verify the generalized Legendre polynomials are derived from the generalized Rodrigues formula.

(iv)

We study of the orthogonality of the generalized Legendre polynomials. To address this problem, the extension of the generalized fractional derivative is needed in the sense of the Abu-Shady–Kaabar definition.

This research study is formulated as the GDFD, with its essential properties exposed in Section 2. In Section 3, the series solutions of a generalized fractional Legendre differential equation are obtained via the fractional power series technique. Interesting results on generalized Legendre polynomials are introduced in Section 4. In particular, the Rodrigues formula, some recurrence relations, and the orthogonality property for these generalized polynomials are investigated. This research study ends by addressing the representation of a function $f\in L_{\alpha }^2\left(\left[-1,1\right]\right)$ in a series of generalized Legendre polynomials. Some conclusions are drawn in Section 5.

2. Preliminaries

Definition 1. 

For function $h:[0,\infty )\to ℝ$, the GDFD of order $0<\alpha \le 1$, of $h$ at $t>0$ is written as [15],

$$D^{GDFD}h\left(t\right)=\underset{\epsilon \to 0}{\mathit{lim}}\frac{h\left(t+\left(\raisebox{1ex}{$\mathsf{\Gamma }\left(\beta \right)$}\!\left/ \!\raisebox{-1ex}{$\mathsf{\Gamma }\left(\beta -\alpha +1\right)$}\right.\right)\epsilon t^{1-\alpha }\right)-h\left(t\right)}{\epsilon } , \beta >-1,\beta \in R^{+}.$$

(1)

Note that the constraint $\beta >-1$ is provided in the above definition to make sure that the Gamma function is well-defined positive and finite, because it is well-known that this special function is defined for all complex numbers except for the non-positive integers. Otherwise, if $\beta$ is less than or equal to $-1$, the Gamma function will not hold, and it can be infinite which is not suitable for the definition of the above fractional derivative.

If $h$ is $\alpha$-differentiable ($\alpha$-Diff) in some $\left(0,a\right)$, $a>0$, and $\underset{t\to 0^{+}}{\mathit{lim}}{D}^{GDFD }h\left(t\right)$ exists, then it is expressed as

$$D^{GDFD}h\left(0\right)=\underset{t\to 0^{+}}{\mathit{lim}}{D}^{GDFD }h\left(t\right).$$

(2)

Theorem 1 

([15]). Let $0<\alpha \le 1,$ $\beta >-1,\beta \in {ℝ}^{+}$ and let $h,w$ be $\alpha$-Diff at a point $t>0$. Then, we have

(i) 

$D^{GDFD }\left[ah+bw\right]=a D^{GDFD }\left[h\right]+b D^{GDFD }\left[w\right]$, $\forall a,b\in ℝ$;

(ii) 

$D^{GDFD }\left[t^m\right]=\frac{m\mathsf{\Gamma }\left(m\right)}{\mathsf{\Gamma }\left(m-\alpha +1\right)}{t}^{m-\alpha }$, $\forall m\in ℝ$;

(iii) 

$D^{GDFD }\left[\mu \right]=0$, $\forall$ constant function $h\left(t\right)=\mu$;

(iv) 

$D^{GDFD }\left[hw\right]=h{D}^{GDFD }\left[w\right]+w{D}^{GDFD }\left[h\right]$;

(v) 

$D^{GDFD }\left[\left(\frac{h}{w}\right)\right]=\frac{w{D}^{GDFD }\left[h\right]-h{D}^{GDFD }\left[w\right]}{w^2}$.

(vi) 

If, additionally, $h$ is differentiable function, then

$$D^{GDFD}h\left(t\right)=\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{1-\alpha }\frac{dh\left(t\right)}{dt}.$$

The generalized $\alpha$-derivative of certain functions via GDFD is

(i) 

$D^{GDFD }\left[1\right]=0$;

(ii) 

$D^{GDFD }\left[sin\left(kt\right)\right]=\frac{k\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{1-\alpha }cos\left(kt\right)$;

(iii) 

$D^{GDFD }\left[cos\left(kt\right)\right]=-\frac{k\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{1-\alpha }sin\left(kt\right)$;

(iv) 

$D^{GDFD }\left[e^{kt}\right]=\frac{k\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{1-\alpha }{e}^{kt}$.

In addition, the generalized $\alpha$-derivative of the following functions are highlighted as follows:

(i) 

$D^{GDFD }\left[\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right]=1$;

(ii) 

$D^{GDFD }\left[sin\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right)\right]=cos\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right)$;

(iii) 

$D^{GDFD }\left[cos\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right)\right]=-sin\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right)$;

(iv) 

$D^{GDFD }\left[e^{\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right)}\right]=e^{\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{t}^{\alpha }\right)}$.

Theorem 2 

(Chain Rule [16]). Let $0<\alpha \le 1$, $\beta >-1,\beta \in {ℝ}^{+}$, $w$ is generalized $\alpha$-Diff at $t>0,$ and $h$ is differentiable at $w\left(t\right)$, then

$$D^{GDFD }\left[h ○ w\right]\left(t\right)=h^{\prime }\left(w\left(t\right)\right)D^{GDFD }w\left(t\right).$$

(3)

Remark 1 

([16]).  Using the fact that differentiability implies generalized $\alpha$-differentiability and by assuming $w\left(t\right)>0$, Equation (3) can be re-written as follows:

$$D^{GDFD }\left[h ○ w\right]\left(t\right)=\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}w{\left(t\right)}^{\alpha -1}{D}^{GDFD }h\left(w\left(t\right)\right)D^{GDFD }w\left(t\right).$$

(4)

Definition 2 

([15]). $I_{\alpha }^a\left(h\right)\left(t\right)=\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{{\displaystyle \int }}_a^t\frac{h\left(x\right)}{x^{1-\alpha }}dx$, where this integral is basically the usual Riemann improper integral, and $\alpha \in (0,1]$.

From Definition 2, we get the following:

Theorem 3 

([15]). $D^{GDFD }{I}_{\alpha }^a\left(h\right)\left(t\right)=h\left(t\right)$, for $t\ge a$, where $h$ is any continuous function in the domain of $I_{\alpha }$.

Theorem 4 

([17]). Let $a>0$, $\alpha \in (0, 1]$ and $h$ be a continuous real-valued function on interval $\left[a,b\right]$. Let $W$ be any real-valued function with the property $D^{GDFD }\left(W\right)\left(t\right)=h\left(t\right)$ for all $t\in \left[a,b\right]$. Then,

$$\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{{\displaystyle \int }}_a^b\frac{h\left(t\right)}{t^{1-\alpha }}dt=W\left(b\right)-W\left(a\right).$$

(5)

Now, let us consider the most general sequential linear homogeneous generalized fractional differential equation, given by

$${ }^{\left(n\right)}{D}^{GDFD}y\left(t\right)+a_{n-1}{\left(t\right)}^{\left(n-1\right)}{D}^{GDFD}y\left(t\right)+\dots +a_1{\left(t\right)}^{\left(1\right)}{D}^{GDFD}y\left(t\right)+a_0\left(t\right)y\left(t\right)=0.$$

(6)

Assume that $\alpha \in \left(0, 1\right]$ and $h\left(t\right)$ be a real function defined on $\left[0, a\right]$. Here, h(t) is said to be a generalized $\alpha$-analytic at 0 if h(t) can be written as follows:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{\infty }{c}_k{t}^{k\alpha } \left(c_k\in ℝ\right),$$

$\forall t \in \left[0, r\right),$ and r is the radius of convergence of this fractional power series.

Definition 3 

([17]). Let $\alpha \in (0, 1]$ and the functions $a_k\left(t\right)$ be a generalized $\alpha$-analytic at $t=0$, for $k=0,1,2,\dots , n-1$. In this case, the point $t=0$ is said to be a generalized $\alpha$-ordinary point of Equation (6).

Theorem 5 

([17]). Let $\alpha \in (0, 1]$ and $c_0,c_1\in ℝ$. If $t=0$ is a generalized $\alpha$-ordinary point of the fractional differential equation:

$${ }^{\left(2\right)}{D}^{GDFD}y\left(t\right)+a{\left(t\right)}^{\left(1\right)}{D}^{GDFD}y\left(t\right)+b\left(t\right)y\left(t\right)=0,$$

(7)

then there is a solution of the above equation as

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{\infty }{c}_k{t}^{k\alpha },$$

(8)

for $t\in [0, r)$ with $r=min\left\{r_1,r_2\right\}$ and initial conditions $y\left(0\right)=c_0$ and ${ }^{\left(1\right)}{D}^{GDFD}y\left(0\right)=\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha c_1$, where $r_1$ and $r_2$ are the radii of convergence of $a\left(t\right)$ and $b\left(t\right)$, respectively.

3. Generalized Fractional Legendre Equation

Consider the following generalized fractional Legendre differential equation:

$${\left(1-t^{2\alpha }\right)}^{\left(2\right)}{D}^{GDFD}y\left(t\right)-\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t^{\alpha }}^{\left(1\right)}{D}^{GDFD}y\left(t\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}p\left(p+1\right)y\left(t\right)=0$$

(9)

where $\alpha \in (0, 1]$, $t>0$, and $p$ is a positive integer. If $\alpha =\beta =1$, then Equation (9) becomes the classical Legendre differential equation [20]. Since $t=0$ is an ordinary point of Equation (9), we use the generalized fractional power series method and propose the following solution:

$$y\left(t\right)={\displaystyle \sum }_{k=0}^{\infty }{c}_k{t}^{k\alpha }.$$

(10)

By substituting (10) and its generalized fractional derivatives in Equation (9), we have

$$\begin{gathered} 2c_2+3.2 c_3{t}^{\alpha }-2c_1{t}^{\alpha }+p\left(p+1\right)c_0+p\left(p+1\right)c_1{t}^{\alpha } \\ +{\displaystyle \sum }_{k=2}^{\infty }\left[\left(k+2\right)\left(k+1\right)c_{k+2}+\left(p+k+1\right)\left(p-k\right)c_k\right]t^{k\alpha }=0. \end{gathered}$$

Note that the recurrence formula above can be written as

$$c_{k+2}=-\frac{\left(p+k+1\right)\left(p-k\right)}{\left(k+2\right)\left(k+1\right)}{c}_k ,k\ge 0 .$$

Hence, we obtain the following:

$$\begin{array}{l}{c}_2=-\frac{p\left(p+1\right)}{2!}{c}_0.\\ c_3=-\frac{\left(p+2\right)\left(p-1\right)}{3!}{c}_1.\\ c_4=-\frac{\left(p+3\right)\left(p-2\right)}{4.3}{c}_2=\frac{\left(p+3\right)\left(p+1\right)p\left(p-2\right)}{4!}{c}_0.\\ c_5=-\frac{\left(p+4\right)\left(p-3\right)}{5.4}{c}_3=\frac{\left(p+4\right)\left(p+2\right)\left(p-1\right)\left(p-3\right)}{5!}{c}_1.\\ \dots \dots \dots \dots \dots \dots \end{array}$$

In general, we can write:

$$\begin{gathered} c_{2k}={\left(-1\right)}^k\frac{\left(p+2k-1\right)\left(p+2k-3\right)\dots \left(p+1\right)p\left(p-2\right)\dots \left(p-2k+2\right)}{\left(2k\right)!}{c}_0. \\ {c}_{2k+1}={\left(-1\right)}^k\frac{\left(p+2k\right)\left(p+2k-2\right)\dots \left(p+2\right)\left(p-1\right)\left(p-3\right)\dots \left(p-2k+1\right)}{\left(2k+1\right)!}{c}_1. \end{gathered}$$

for $k\ge 1$.

As a result, we have the following general solution of Equation (9):

$$\begin{gathered} y\left(t\right)=c_0\left(1+{\displaystyle \sum }_{k=1}^{\infty }{\left(-1\right)}^k\frac{\left(p+2k-1\right)\left(p+2k-3\right)\dots \left(p+1\right)p\left(p-2\right)\dots \left(p-2k+2\right)}{\left(2k\right)!}{t}^{2k\alpha }\right) \\ +c_1\left(t^{\alpha }+{\displaystyle \sum }_{k=1}^{\infty }{\left(-1\right)}^k\frac{\left(p+2k\right)\left(p+2k-2\right)\dots \left(p+2\right)\left(p-1\right)\left(p-3\right)\dots \left(p-2k+1\right)}{\left(2k+1\right)!}{t}^{\left(2k+1\right)\alpha }\right). \end{gathered}$$

Remark 2. 

Note that the above equation can be written in the following form:

$$y\left(t\right)=c_0{y}_1\left(t\right)+c_1{y}_2\left(t\right),$$

where

$$y_1\left(t\right)=1+{\displaystyle \sum }_{k=1}^{\infty }{\left(-1\right)}^k\frac{\left(p+2k-1\right)\left(p+2k-3\right)\dots \left(p+1\right)p\left(p-2\right)\dots \left(p-2k+2\right)}{\left(2k\right)!}{t}^{2k\alpha }.$$

(11)

$$y_2\left(t\right)=t^{\alpha }+{\displaystyle \sum }_{k=1}^{\infty }{\left(-1\right)}^k\frac{\left(p+2k\right)\left(p+2k-2\right)\dots \left(p+2\right)\left(p-1\right)\left(p-3\right)\dots \left(p-2k+1\right)}{\left(2k+1\right)!}{t}^{\left(2k+1\right)\alpha }.$$

(12)

To find the radii of convergence of the series (11) and series (12), we use ratio test as follows:

$$\begin{array}{c}\underset{k\to \infty }{\mathit{lim}}\left|\frac{\left(2k + 2\right)!\left(p + 2k - 1\right)\left(p + 2k - 3\right)\dots \left(p + 1\right)p\left(p - 2\right)\dots \left(p - 2k + 2\right)}{\left(2k\right)!\left(p + 2k + 1\right)\left(p + 2k - 1\right)\dots \left(p + 1\right)p\left(p - 2\right)\dots \left(p - 2k + 2\right)\left(p - 2k\right)}\right|\\ =\underset{k\to \infty }{\mathit{lim}}\left|\frac{\left(2k + 2\right)\left(2k + 1\right)}{\left(p + 2k + 1\right)\left(p - 2k\right)}\right|=1\\ \underset{k\to \infty }{\mathit{lim}}\left|\frac{\left(2k + 3\right)!\left(p + 2k\right)\left(p + 2k - 2\right)\dots \left(p + 2\right)\left(p - 1\right)\left(p - 3\right)\dots \left(p - 2k + 1\right)}{\left(2k + 1\right)!\left(p + 2k + 2\right)\left(p + 2k\right)\dots \left(p + 2\right)\left(p - 1\right)\left(p - 3\right)\dots \left(p - 2k - 1\right)}\right|\\ =\underset{k\to \infty }{\mathit{lim}}\left|\frac{\left(2k + 3\right)\left(2k + 2\right)}{\left(p + 2k + 2\right)\left(p - 2k - 1\right)}\right|=1\end{array}$$

Thus, the radius of convergence is $r=1$ in both cases, and in that case $y\left(t\right)$ converges for $t\in [0, 1)$.

Remark 3. 

Let us now consider $p$ to be a non-negative integer.

• If $p$ is even, that is $p=2k$, then $y_1\left(t\right)$ is a polynomial of degree $p\alpha$, since the factor $\left(p-2k\right)$ is obtained from the coefficient of $t^{2k\alpha }$. For example, if $p=2$, $y_1\left(t\right)=1-3t^{2\alpha }$; if $p=4$, $y_1\left(t\right)=1-10t^{2\alpha }+\frac{35}{3}{t}^{2\alpha }$; if $p=6$, $y_1\left(t\right)=1-21t^{2\alpha }+63t^{4\alpha }-\frac{694}{15}{t}^{6\alpha }$, etc.
• If $p$ is odd, that is $p=2k+1$, then $y_2\left(t\right)$ is a polynomial of degree $p\alpha$, since the factor $\left(p-2k-1\right)$ is obtained from the coefficient of $t^{\left(2k+1\right)\alpha }$. For example, if $p=1$, $y_2\left(t\right)=t^{\alpha }$; if $p=3$, $y_2\left(t\right)=t^{\alpha }+\frac{5}{3}{t}^{3\alpha }$; if $p=5$, $y_2\left(t\right)=t^{\alpha }-\frac{14}{3}{t}^{3\alpha }+\frac{21}{5}{t}^{5\alpha }$, etc.

In short, if $p$ is a non-negative integer, there is a polynomial solution to the generalized Legendre equation. These polynomial solutions, when certain additional conditions are imposed on them, form a set of polynomials, which we will call generalized Legendre polynomials, of which it is very interesting to study some of their properties.

4. Generalized Legendre Polynomials and Its Properties

In this section, we establish important results of the Legendre polynomials using the Abu-Shady–Kaabar fractional derivative. Specifically, we construct the Rodrigues formula through which we guarantee the existence and uniqueness of these fractional polynomials. Likewise, we develop important recurrence relations involving these polynomials. The study of the orthogonality of fractional Legendre polynomials will also be the goal of special attention. Finally, we investigate the representation of a function $f\in L_{\alpha }^2\left(\left[-1, 1\right]\right)$ in a series of generalized Legendre polynomials.

4.1. Generalized Rodrigues Formula

Definition 4. 

Let $P_n^{\alpha }\left(t\right)$ be a polynomial of degree $n\alpha$ that satisfies generalized fractional Legendre differential equation with p = n such that $P_n^{\alpha }\left(1\right)=1$. We will call $P_n^{\alpha }\left(t\right)$ a generalized Legendre polynomial of degree $n\alpha$.

Now, we are going to see that there is a formula that encompasses all the generalized Legendre polynomials. To do this, we establish the following previous result:

Lemma 1. 

Let u$\left(t\right)={ }^{\left(n\right)}{D}^{GDFD}{\left(t^{2\alpha }-1\right)}^n$. Then, u$\left(t\right)$ is a polynomial solution of generalized fractional Legendre differential equation (with $p=n$)

$${\left(1-t^{2\alpha }\right)}^{\left(2\right)}{D}^{GDFD}y\left(t\right)-\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t^{\alpha }}^{\left(1\right)}{D}^{GDFD}y\left(t\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}n\left(n+1\right)y\left(t\right)=0.$$

(13)

Proof. 

If we take $v\left(t\right)={\left(t^{2\alpha }-1\right)}^n$, then

$${ }^{\left(1\right)}{D}^{GDFD}v\left(t\right)=\frac{2\alpha n\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{\left(t^{2\alpha }-1\right)}^{n-1}.$$

(14)

Multiplying by $\left(t^{2\alpha }-1\right)$, we get:

$${\left(t^{2\alpha }-1\right)}^{\left(1\right)}{D}^{GDFD}v\left(t\right)=\frac{2\alpha n\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }v\left(t\right).$$

If we compute the generalized $\alpha$-derivative of the above equation, we have:

$$\left(t^{2\alpha }-1\right){}_{ }{}^{\left(2\right)}D{ }^{GDFD}v\left(t\right)-\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD}v\left(t\right)-2n{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}v\left(t\right)=0.$$

In view of this last equation, we propose the following expression involving the generalized derivative of order $k\alpha$ of the function $v\left(t\right)$:

$$\left(t^{2\alpha }-1\right){}_{ }{}^{\left(k\right)}D{ }^{GDFD}v\left(t\right)-2\left(k-1-n\right)\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(k-1\right)}D{ }^{GDFD}v\left(t\right)+\left(k-1\right)\left(k-2n-2\right){\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^2{}_{ }{}^{\left(k-2\right)}D{ }^{GDFD}v\left(t\right)=0.$$

(15)

Reasoning inductively, we are going to prove the validity of this statement. Indeed, for $k=1$, it is true. Assuming the statement for $k$ is valid and computing the generalized $\alpha$-derivative of this expression, we obtain

$$\left(t^{2\alpha }-1\right){}_{ }{}^{\left(k\right)}D{ }^{GDFD}v\left(t\right)-2\left(k-n\right)\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(k-1\right)}D{ }^{GDFD}v\left(t\right)+k\left(k-2n-1\right){\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^2{}_{ }{}^{\left(k-2\right)}D{ }^{GDFD}v\left(t\right)=0.$$

Finally, it can be concluded that Equation (15) is valid for all $k\in N$.

If we now take $k=n+2$ in (15), we get

$$\left(t^{2\alpha }-1\right){}_{ }{}^{\left(n+2\right)}D{ }^{GDFD}v\left(t\right)-2\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(n+1\right)}D{ }^{GDFD}v\left(t\right)+n\left(n+1\right){\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^2{}_{ }{}^{\left(n\right)}D{ }^{GDFD}v\left(t\right)=0.$$

But since ${ }^{\left(n\right)}{D}^{GDFD}v\left(t\right)=u\left(t\right)$, the above equation can be written as

$$\left(t^{2\alpha }-1\right){}_{ }{}^{\left(2\right)}D{ }^{GDFD}v\left(t\right)-2\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD}v\left(t\right)+n\left(n+1\right){\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}v\left(t\right)=0,$$

which completes the proof. □

Remark 4. 

It is easy to show that the polynomial u$\left(t\right)$ of degree $n\alpha$ from the above lemma satisfies u$\left(1\right)=1$. In fact, since u$\left(t\right)$ can be expressed as

$$\begin{array}{l}u\left(t\right)={ }^{\left(n\right)}{D}^{GDFD}{\left(t^{2\alpha }-1\right)}^n={ }^{\left(n\right)}{D}^{GDFD}\left[{\left(t^{\alpha }-1\right)}^n{\left(t^{\alpha }+1\right)}^n\right]\\ ={\displaystyle {\displaystyle \sum }_{j=0}^n}{}_{ }{}^{\left(j\right)}D{ }^{GDFD}{{\left(t^{\alpha }-1\right)}^n}^{\left(n-j\right)}{D}^{GDFD}{\left(t^{\alpha }+1\right)}^n={ }^{\left(n\right)}{D}^{GDFD}{\left(t^{\alpha }-1\right)}^n·{\left(t^{\alpha }+1\right)}^n\\ +{\displaystyle {\displaystyle \sum }_{j=0}^{n-1}}{}_{ }{}^{\left(j\right)}D{ }^{GDFD}{{\left(t^{\alpha }-1\right)}^n}^{\left(n-j\right)}{D}^{GDFD}{\left(t^{\alpha }+1\right)}^n.\end{array}$$

and, furthermore, that

$${ }^{\left(n\right)}{D}^{GDFD}{\left(t^{\alpha }-1\right)}^n={\left(\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\right)}^{n}n!,$$

can be written as:

$$u\left(t\right)={\left(\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\right)}^{n}n!{\left(t^{\alpha }+1\right)}^n+{\displaystyle \sum }_{j=0}^{n-1}{}_{ }{}^{\left(j\right)}D{ }^{GDFD }{\left(t^{\alpha }-1\right)}^n{}_{ }{}^{\left(n-j\right)}D{ }^{GDFD }{\left(t^{\alpha }+1\right)}^n.$$

Finally, it follows that $u\left(t\right)={\left(\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\right)}^{n}n!$.

Remark 5. 

The fractional polynomials, given by

$$P_n^{\alpha }\left(t\right)={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }{\left(t^{2\alpha }-1\right)}^n$$

(16)

satisfy the generalized fractional Legendre differential equation (with $p=n$) and also $P_n^{\alpha }\left(1\right)=1$. Equation (16) constitutes a generating formula for Legendre’s fractional polynomials, which we call the generalized Rodrigues formula. Through this formula, we guarantee the existence of these fractional polynomials, but it also allows us to establish an argument about the uniqueness of these polynomials, as we will study it next.

Lemma 2. 

The unique polynomial solution $P^{\alpha }\left(t\right)$ of generalized fractional Legendre differential equation with $p=n$:

$$\left(1-t^{2\alpha }\right){}_{ }{}^{\left(2\right)}D{ }^{GDFD }y\left(t\right)-\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }y\left(t\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}n\left(n+1\right)y\left(t\right)=0,$$

such that $P^{\alpha }\left(1\right)=0$ is the solution $P^{\alpha }\left(t\right)\equiv 0$.

Proof. 

Let $P^{\alpha }\left(t\right)$ be a polynomial solution of generalized fractional Legendre differential equation. Then, there exist real constants $k_1$ and $k_2$ such that

$$P^{\alpha }\left(t\right)=k_1{y}_1\left(t\right)+k_2{y}_2\left(t\right),$$

where $y_1\left(t\right)$ and $y_2\left(t\right)$ are the fractional series that appear in the previously established general solution: $y\left(t\right)=c_0{y}_1\left(t\right)+c_1{y}_2\left(t\right).$

If $p=n,$ we know that either $y_1\left(t\right)$ is a polynomial solution (if $n$ is even) or $y_2\left(t\right)$ is (if $n$ is odd).

If we assume that $n$ is even, then

$$P^{\alpha }\left(t\right)-k_1{y}_1\left(t\right)=k_2{y}_2\left(t\right)$$

where the LHS of the above equation is a polynomial solution of generalized fractional Legendre equation since $P^{\alpha }\left(t\right)$ is. However, note that $y_2\left(t\right)$ is not a polynomial (it is an infinite power series), so it easily follows that $k_2=0$. A similar reasoning for n odd allows us to finally conclude that if $P^{\alpha }\left(t\right)$ is a polynomial solution of generalized fractional Legendre equation with $p=n$, then there exist real constants $k_1$ or $k_2$, such that $P^{\alpha }\left(t\right)=k_1{y}_1\left(t\right)$ or $P^{\alpha }\left(t\right)=k_2{y}_2\left(t\right)$.

In particular, $P_n^{\alpha }\left(t\right)$ given by the generalized Rodrigues formula is such a solution. Then, we have that $P_n^{\alpha }\left(t\right)=k_1{y}_1\left(t\right)$ or $P_n^{\alpha }\left(t\right)=k_2{y}_2\left(t\right)$.

Since $P_n^{\alpha }\left(1\right)=1$, then we conclude that $y_1\left(1\right)\ne 0$ or $y_2\left(1\right)\ne 0$. Therefore, if $P^{\alpha }\left(t\right)$ is a polynomial solution of the generalized fractional Legendre equation with $p=n$, such that $P^{\alpha }\left(1\right)=0$, since $P^{\alpha }\left(t\right)=k_1{y}_1\left(t\right)$ or $P^{\alpha }\left(t\right)=k_2{y}_2\left(t\right)$, where $y_1\left(t\right)\ne 0$ and $y_2\left(t\right)\ne 0$, then we must have $k_1=0$ or $k_2=0$, respectively.

In any case, $P^{\alpha }\left(t\right)\equiv 0$. □

Remark 6. 

The uniqueness of the fractional Legendre polynomials easily follows from the fact that, if in addition to those given by the Rodrigues formula, there were others, say ${\tilde{P}}_n^{\alpha }\left(t\right)$ we would have that $P^{\alpha }\left(t\right)=P_n^{\alpha }\left(t\right)-{\tilde{P}}_n^{\alpha }\left(t\right)$ which would be a polynomial solution of the generalized fractional Legendre equation such that $P^{\alpha }\left(1\right)=P_n^{\alpha }\left(1\right)-{\tilde{P}}_n^{\alpha }\left(1\right)=0$. Then, $P^{\alpha }\left(t\right)\equiv 0$, that is, $P_n^{\alpha }\left(t\right)\equiv {\tilde{P}}_n^{\alpha }\left(t\right)$.

We can now state that the fractional Legendre polynomials are

$$P_n^{\alpha }\left(t\right)={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }{\left(t^{2\alpha }-1\right)}^n.$$

Some of these polynomials are

$$\begin{gathered} P_0^{\alpha }\left(t\right)=1. \\ {P}_1^{\alpha }\left(t\right)=t^{\alpha }. \\ {P}_2^{\alpha }\left(t\right)=\frac{3}{2}{t}^{2\alpha }-\frac{1}{2}. \\ {P}_3^{\alpha }\left(t\right)=\frac{5}{2}{t}^{3\alpha }-\frac{3}{2}{t}^{\alpha } \end{gathered}$$

and so on.

Remark 7. 

Finally, we will show that the coefficient of$t^{n\alpha }$ in $P_n^{\alpha }\left(t\right)$ is $\frac{\left(2n\right)!}{{\left(n!\right)}^2{2}^n}$.

So we have

$$\begin{gathered} P_n^{\alpha }\left(t\right)={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }{\left(t^{2\alpha }-1\right)}^n \\ ={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left[{\displaystyle \sum }_{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right){\left(t^{2\alpha }\right)}^{n-j}{\left(-1\right)}^j\right] \\ ={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left[t^{2n\alpha }+{\displaystyle \sum }_{j=1}^n\left(\begin{array}{c}n\\ j\end{array}\right){\left(t^{2\alpha }\right)}^{n-j}{\left(-1\right)}^j\right] \end{gathered}$$

Notice that in the expression

$${}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left[{\displaystyle \sum }_{j=1}^n\left(\begin{array}{c}n\\ j\end{array}\right){\left(t^{2\alpha }\right)}^{n-j}{\left(-1\right)}^j\right],$$

only the powers of $t^{\alpha }$ less than $n$ appear. Thus, the coefficient of $t^{n\alpha }$ in $P_n^{\alpha }\left(t\right)$ comes only from the generalized derivative of order $n\alpha$ of $t^{2n\alpha }$, which is

$${}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left(t^{2n\alpha }\right)={\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\alpha \mathsf{\Gamma }\left(\beta \right)\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\right)}^n\left(2n\right)\left(2n-1\right)\left(2n-2\right)\dots \left(n+1\right)t^{n\alpha }.$$

Then, the coefficient of $t^{n\alpha }$ in $P_n^{\alpha }\left(t\right)$ is

$$\frac{\left(2n\right)\left(2n-1\right)\left(2n-2\right)\dots \left(n+1\right)}{n!2^n}.$$

Finally, if we multiply numerator and denominator by $n!$, the result follows.

4.2. Some Important Recurrence Relations

We use the generalized Rodrigues formula to obtain several very useful recurrence relations. Note that

$$\begin{array}{l}{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)={}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left[{\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^{n+1}\frac{1}{\left(n+1\right)!}{}_{ }{}^{\left(n+1\right)}D{ }^{GDFD }{\left(t^{2\alpha }-1\right)}^{n+1}\right]\\ ={}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left[{\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left(t^{\alpha }{\left(t^{2\alpha }-1\right)}^n\right)\right]\\ ={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n+1\right)}D{ }^{GDFD }\left(t^{\alpha }{\left(t^{2\alpha }-1\right)}^n\right).\end{array}$$

(17)

Therefore, by computing the generalized $\alpha$-derivative of the term in parentheses, we have

$$\begin{gathered} {}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left[{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(t^{\alpha }{\left(t^{2\alpha }-1\right)}^n\right)\right] \\ ={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left[\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\left({\left(t^{2\alpha }-1\right)}^n+2n{t}^{2\alpha }{\left(t^{2\alpha }-1\right)}^{n-1}\right)\right] \\ ={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left[\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\left(\left(2n+1\right){\left(t^{2\alpha }-1\right)}^n+2n{\left(t^{2\alpha }-1\right)}^{n-1}\right)\right] \\ =\frac{\left(2n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{P}_n^{\alpha }\left(t\right)+{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n-1}^{\alpha }\left(t\right)\right), n=1,2,3,\dots \end{gathered}$$

(18)

From Equation (17), another recurrence relation can be obtained, if we consider the effect of repeatedly differentiating a product of the form $t^{2\alpha }f\left(t\right)$.

Note that

$$\begin{gathered} {}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(t^{\alpha }f\left(t\right)\right)=t^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(f\left(t\right)\right)+\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}f\left(t\right), \\ {}_{ }{}^{\left(2\right)}D{ }^{GDFD }\left(t^{\alpha }f\left(t\right)\right)=t^{\alpha }{}_{ }{}^{\left(2\right)}D{ }^{GDFD }\left(f\left(t\right)\right)+\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(f\left(t\right)\right), \end{gathered}$$

and in general

$${}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left(t^{\alpha }f\left(t\right)\right)=t^{\alpha }{}_{ }{}^{\left(n\right)}D{ }^{GDFD }\left(f\left(t\right)\right)+\frac{n\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{}_{ }{}^{\left(n-1\right)}D{ }^{GDFD }\left(f\left(t\right)\right).$$

(19)

Applying Equation (19) to the parenthetical expression of Equation (17), we obtain

$$\begin{gathered} {}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)={\left(\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{2\alpha \mathsf{\Gamma }\left(\beta \right)}\right)}^n\frac{1}{n!}\left[t^{\alpha }{}_{ }{}^{\left(n+1\right)}D{ }^{GDFD }{\left(t^{2\alpha }-1\right)}^n+\frac{\left(n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{}_{ }{}^{\left(n\right)}D{ }^{GDFD }{\left(t^{2\alpha }-1\right)}^n\right] \\ =t^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_n^{\alpha }\left(t\right)\right)+\frac{\left(n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{P}_n^{\alpha }\left(t\right), n=1,2,3,\dots \end{gathered}$$

(20)

In summary, we have established the following identities:

$$\frac{\left(n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{P}_n^{\alpha }\left(t\right)={}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)-t^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_n^{\alpha }\left(t\right)\right).$$

(21)

$$\frac{\left(2n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{P}_n^{\alpha }\left(t\right)={}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)-{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n-1}^{\alpha }\left(t\right)\right),$$

(22)

for $n=1,2,3\dots$

Subtracting the first identity from the second, we have

$$n{P}_n^{\alpha }\left(t\right)=t^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_n^{\alpha }\left(t\right)\right)-{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n-1}^{\alpha }\left(t\right)\right),$$

(23)

for $n=1,2,3\dots$

If we now combine Equations (21)–(23), we obtain

$$\begin{gathered} \frac{\left(n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{P}_{n+1}^{\alpha }\left(t\right)-\frac{\left(2n+1\right)\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{P}_n^{\alpha }\left(t\right)+\frac{n\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{P}_{n-1}^{\alpha }\left(t\right) \\ =t^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)-{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_n^{\alpha }\left(t\right)\right)-t^{\alpha }\left({}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\left(t\right)\right)-{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n-1}^{\alpha }\left(t\right)\right)\right) \\ +{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_n^{\alpha }\left(t\right)\right)-t^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n-1}^{\alpha }\left(t\right)\right). \end{gathered}$$

Thus, by eliminating all derivatives, we obtain the following identity:

$$\left(n+1\right)P_{n+1}^{\alpha }\left(t\right)-\left(2n+1\right)t^{\alpha }{P}_n^{\alpha }\left(t\right)+n{P}_{n-1}^{\alpha }\left(t\right)=0,$$

(24)

for $n=1,2,3\dots$

The second-order homogeneous linear difference equation in Equation (24) can be used to generate all fractional Legendre polynomials if $P_0^{\alpha }\left(t\right)$ and $P_1^{\alpha }\left(t\right)$ are given.

We will illustrate this iteration technique in the following example:

Example 1. 

If take $P_0^{\alpha }\left(t\right)=1$ and $P_1^{\alpha }\left(t\right)=t^{\alpha }$, we are going to calculate $P_2^{\alpha }\left(t\right)$, $P_3^{\alpha }$, $P_4^{\alpha }$, $P_5^{\alpha }$, $P_6^{\alpha }$, $P_7^{\alpha }$, $P_8^{\alpha }$ and $P_9^{\alpha }\left(t\right)$.

Rewriting identity (24)

$$P_{n+1}^{\alpha }\left(t\right)=\frac{\left(2n+1\right)t^{\alpha }{P}_n^{\alpha }\left(t\right)+n{P}_{n-1}^{\alpha }\left(t\right)}{\left(n+1\right)},$$

we have

$$\begin{gathered} P_2^{\alpha }\left(t\right)=\frac{3t^{\alpha }{P}_1^{\alpha }\left(t\right)+P_0^{\alpha }\left(t\right)}{2}=\frac{1}{2}\left(3t^{2\alpha }-1\right). \\ {P}_3^{\alpha }\left(t\right)=\frac{5t^{\alpha }{P}_2^{\alpha }\left(t\right)+2P_1^{\alpha }\left(t\right)}{3}=\frac{1}{2}\left(5t^{3\alpha }-3t^{\alpha }\right). \\ {P}_4^{\alpha }\left(t\right)=\frac{1}{8}\left(35t^{4\alpha }-30t^{2\alpha }+3\right). \\ {P}_5^{\alpha }\left(t\right)=\frac{1}{8}\left(65t^{5\alpha }-70t^{3\alpha }+15t^{\alpha }\right). \\ {P}_6^{\alpha }\left(t\right)=\frac{1}{16}\left(231t^{6\alpha }-315t^{4\alpha }+105t^{2\alpha }-5\right). \\ {P}_7^{\alpha }\left(t\right)=\frac{1}{16}\left(429t^{7\alpha }-693t^{5\alpha }+315t^{3\alpha }-35t^{\alpha }\right). \\ {P}_8^{\alpha }\left(t\right)=\frac{1}{128}\left(6435t^{8\alpha }-12012t^{6\alpha }+6930t^{4\alpha }-1260t^{2\alpha }+36\right). \\ {P}_9^{\alpha }\left(t\right)=\frac{1}{128}\left(12155t^{9\alpha }-25740t^{7\alpha }+18018t^{5\alpha }-4620t^{3\alpha }+315t^{\alpha }\right), \end{gathered}$$

and so on.

4.3. Orthogonality of Generalized Legendre Polynomials

In this subsection, we will study another important property of generalized Legendre polynomials and their orthogonality. First, we recall this property for classical Legendre polynomials.

Theorem 6 

([20]). Let ${\left\{P_n\left(t\right)\right\}}_{n=0}^{\infty }$ be a set of Legendre polynomials of degree n. Then, it is verified that

$${\displaystyle {{\displaystyle \int }}_{-1}^1}{P}_n\left(t\right)P_m\left(t\right)dt=\{\begin{array}{cc}0& if n\ne m\\ \frac{2}{2n+1}& if n=m\end{array} .$$

(25)

Remark 8. 

Note that the norm of the polynomials $P_n\left(t\right)$ is $\sqrt{\frac{2}{2n+1}}$.

If we want to study the orthogonality of generalized Legendre polynomials, as in the classical case, over the interval [−1, 1], then we must extend Definition 1 to include negative values of t. To avoid the trouble of being undefined on $\left[-1, 0\right]$, we assume that $\alpha$ is of the form $\frac{1}{k}$, with $k$ as an odd natural number. So throughout this subsection, we assume $\alpha =\frac{1}{2j+1}$, where $j$ is any natural number. In such a case, $t^{1-\alpha }$ and $t^{n\alpha }$ are defined for all $t\in ℝ$, with $n\in \mathbb{N}$.

Definition 5. 

For function $h:ℝ\to ℝ$, the GDFD of order $0<\alpha =\frac{1}{2j+1}\le 1$, of $h$ at $t\ne 0$ is written same as in Definition 1. If $t=0$, then we define

$$D^{GDFD }h\left(0\right)=\underset{t\to 0}{\mathit{lim}}{D}^{GDFD }h\left(t\right).$$

(26)

Remark 9. 

From the above definition, the following statements follow:

(i) 

For the case of the generalized polynomials, we have

$$D^{GDFD }\left[t^{n\alpha }\right]=\frac{n\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\left(n-1\right)\alpha },\forall t\in ℝ.$$

(ii) 

We can also write the following generalized $\alpha -$ integral function of ℎ where the lower bound of integration is $-1$:

$$I_{\alpha }^{-1}\left(h\right)\left(1\right)=\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{h\left(t\right)}{t^{1-\alpha }}dt.$$

We are now in position to establish the main results on the orthogonality of the generalized Legendre polynomials.

Theorem 7. 

Let ${\left\{P_n^{\alpha }\left(t\right)\right\}}_{n=0}^{\infty }$ be the collection of generalized Legendre polynomials of degree $n\alpha$. Then, it is verified that

$$I_{\alpha }^{-1}\left(P_n^{\alpha }{P}_m^{\alpha }\right)\left(1\right)=\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{P_n^{\alpha }\left(t\right)P_m^{\alpha }\left(t\right)}{t^{1-\alpha }}dt=0 if n\ne m.$$

(27)

Proof. 

Since $P_n^{\alpha }\left(t\right)$ is a solution to the generalized fractional differential equation

$$\left(1-t^{2\alpha }\right){}_{ }{}^{\left(2\right)}D{ }^{GDFD }y\left(t\right)-\frac{2\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{t}^{\alpha }{}_{ }{}^{\left(1\right)}D{ }^{GDFD }y\left(t\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}n\left(n+1\right)y\left(t\right)=0.$$

Then,

$${}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(\left(1-t^{2\alpha }\right){}_{ }{}^{\left(1\right)}D{ }^{GDFD }{P}_n^{\alpha }\left(t\right)\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}n\left(n+1\right)P_n^{\alpha }\left(t\right)=0.$$

(28)

Similarly, for $P_m^{\alpha }\left(t\right)$, we get

$${}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(\left(1-t^{2\alpha }\right){}_{ }{}^{\left(1\right)}D{ }^{GDFD }{P}_m^{\alpha }\left(t\right)\right)+{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^{2}m\left(m+1\right)P_m^{\alpha }\left(t\right)=0.$$

(29)

Multiplying Equation (28) by $P_m^{\alpha }\left(t\right)$ and Equation (29) by $P_n^{\alpha }\left(t\right)$ and subtracting the resulting equation to get

$$\begin{gathered} {}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(\left(1-t^{2\alpha }\right){}_{ }{}^{\left(1\right)}D{ }^{GDFD }{P}_n^{\alpha }\left(t\right)\right)P_m^{\alpha }\left(t\right)-{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(\left(1-t^{2\alpha }\right){}_{ }{}^{\left(1\right)}D{ }^{GDFD }{P}_m^{\alpha }\left(t\right)\right)P_n^{\alpha }\left(t\right) \\ +{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^2\left(n\left(n+1\right)-m\left(n+1\right)\right)P_n^{\alpha }\left(t\right)P_m^{\alpha }\left(t\right)=0, \end{gathered}$$

or

$$\begin{array}{l}{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(\left(1-t^{2\alpha }\right)\left({}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_n^{\alpha }\left(t\right)\right)P_m^{\alpha }\left(t\right)-P_n^{\alpha }\left(t\right){}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_m^{\alpha }\left(t\right)\right)\right)\right)\\ +{\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}\alpha \right)}^2\left(n-m\right)\left(n+m+1\right)P_n^{\alpha }\left(t\right)P_m^{\alpha }\left(t\right)=0.\end{array}$$

(30)

Now, by applying the generalized integral $I_{\alpha }^{-1}$ to Equation (30), we obtain

$${\left(\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}\alpha \right)}^2\left(n-m\right)\left(n+m+1\right){\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{P_n^{\alpha }\left(t\right)P_m^{\alpha }\left(t\right)}{t^{1-\alpha }}·dt=0.$$

Finally, if $n\ne m$

$$\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{P_n^{\alpha }\left(t\right)P_m^{\alpha }\left(t\right)}{t^{1-\alpha }}·dt=0.$$

□

Once the orthogonality of the generalized Legendre polynomials has been established, we proceed to find the value of its norm through the following result:

Theorem 8. 

For $n=0,1,2,\dots$ we have that

$$I_{\alpha }^{-1}\left({\left(P_n^{\alpha }\right)}^2\right)\left(1\right)=\frac{2}{2n+1}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}.$$

(31)

that is, that the norm of the polynomials $P_n^{\alpha }\left(t\right)$ is $\sqrt{\frac{2}{2n+1}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}}$.

Proof. 

To determine the result of the integral $I_{\alpha }^{-1}\left({\left(P_n^{\alpha }\right)}^2\right)\left(1\right),$ we proceed as follows: First, if in Equation (25), we replace $n$ by $n+1$ and multiply by $\left(2n+1\right)P_n^{\alpha }\left(t\right)$, we obtain

$$n\left(2n+1\right){\left(P_n^{\alpha }\left(t\right)\right)}^2-\left(2n-1\right)\left(2n+1\right)t^{\alpha }{P}_n^{\alpha }\left(t\right)P_{n-1}^{\alpha }\left(t\right)+\left(n-1\right)\left(2n+1\right)P_n^{\alpha }\left(t\right)P_{n-2}^{\alpha }\left(t\right)=0.$$

Subtracting from this last relation, Equation (25) multiplied by $\left(2n-1\right)P_{n-1}^{\alpha }\left(t\right)$, we have

$$n\left(2n+1\right){\left(P_n^{\alpha }\left(t\right)\right)}^2+\left(n-1\right)\left(2n+1\right)P_n^{\alpha }\left(t\right)P_{n-2}^{\alpha }\left(t\right)-\left(n+1\right)\left(2n-1\right)P_{n-1}^{\alpha }\left(t\right)P_{n-2}^{\alpha }\left(t\right)-n\left(2n-1\right){\left(P_{n-1}^{\alpha }\left(t\right)\right)}^2=0,$$

for $n=2,3,4,\dots$

Now, by applying the generalized integral $I_{\alpha }^{-1}$ to above equation, we obtain

$$n\left(2n+1\right)\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_n^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt=n\left(2n-1\right)\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_{n-1}^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt,$$

for $n=2,3,4,\dots$

or

$$\begin{array}{l}\frac{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_n^{\alpha }\left(t\right)\right)}^2}{t^{1 - \alpha }}dt\\ =\frac{2n - 1}{2n + 1}\frac{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_{n-1}^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt\\ =\frac{2n - 3}{2n + 1}\frac{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_{n-2}^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt=\dots \\ =\frac{3}{2n + 1}\frac{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_1^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt=\frac{2}{2n + 1}\frac{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}.\end{array}$$

In addition,

$$\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_0^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt=2\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}=\frac{2}{2·0+1}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)},$$

and

$$\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{{\left(P_1^{\alpha }\left(t\right)\right)}^2}{t^{1-\alpha }}dt=\frac{2}{3}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}=\frac{2}{2·1+1}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}.$$

Finally, our result follows. □

Remark 10. 

From these results, we observe that the generalized Legendre polynomials are not normalized and, furthermore, by joining Theorems 7 and 8, we obtain

$$I_{\alpha }^{-1}\left(P_n^{\alpha }{P}_m^{\alpha }\right)\left(1\right)=\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\mathsf{\Gamma }\left(\beta \right)}{\displaystyle {{\displaystyle \int }}_{-1}^1}\frac{P_n^{\alpha }\left(t\right)P_m^{\alpha }\left(t\right)}{t^{1-\alpha }}dt=\frac{2}{2n+1}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{\delta }_{nm},$$

(32)

where ${\delta }_{nm}$ is the Kronecker delta.

4.4. Generalized Legendre Series

In this subsection, we consider $L_{\alpha }^2\left(\left[-1, 1\right]\right)$, the space of Lebesgue measurable functions $f:\left[-1, 1\right]\to R$ with square $\alpha$-integrable, that is, $I_{\alpha }^{-1}\left(f^2\right)\left(1\right)<+\infty$, where the integral is understood in the Lebesgue sense. In this subsection, it is also assumed that $\alpha =\frac{1}{2j+1}$, where $j$ is any natural number. Now, we are going to approach the representation of a function $f\in L_{\alpha }^2\left(\left[-1, 1\right]\right)$ in a series of generalized Legendre polynomials:

$$f\left(t\right)={\displaystyle \sum }_{n=0}^{\infty }{c}_n{P}_n^{\alpha }\left(t\right).$$

(33)

Multiplying the above equation by $P_m^{\alpha }\left(t\right)$ and computing the generalized $\alpha$-integral from $-1$ to $1$ gives

$$I_{\alpha }^{-1}\left(f{P}_m^{\alpha }\right)\left(1\right)={\displaystyle \sum }_{n=0}^{\infty }{c}_n{I}_{\alpha }^{-1}\left(P_n^{\alpha }{P}_m^{\alpha }\right)\left(1\right)=\frac{2}{2m+1}\frac{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{\alpha \mathsf{\Gamma }\left(\beta \right)}{c}_m.$$

Hence,

$$c_m=\frac{2m+1}{2}\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{I}_{\alpha }^{-1}\left(f{P}_m^{\alpha }\right)\left(1\right), m=0,1,2,\dots .$$

(34)

This development that we have just introduced can be formalized in the following result:

Theorem 9. 

Let $f\in L_{\alpha }^2\left(\left[-1,1\right]\right)$. Then, $f\left(t\right)$ has a generalized Legendre series representation on the interval $\left[-1,1\right]$ of the form

$$f\left(t\right)={\displaystyle \sum }_{n=0}^{\infty }{c}_n{P}_n^{\alpha }\left(t\right)$$

with

$$c_n=\frac{2n+1}{2}\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{I}_{\alpha }^{-1}\left(f{P}_n^{\alpha }\right)\left(1\right), n=0,1,2,\dots .$$

Remark 11. 

For each $n$, the generalized Legendre polynomials $P_0^{\alpha }, P_1^{\alpha },\dots ,P_n^{\alpha }$ form a basis in the space of polynomials of degree $n\alpha$. Thus, if $Q\left(t\right)$ is a polynomial of degree $n\alpha$, then the generalized Legendre series of $Q$ terminates at the order $n$ (i.e., $c_k=0$ for $k>n$). In particular, we have

$$t^{n\alpha }=c_0{P}_0^{\alpha }\left(t\right)+c_1{P}_1^{\alpha }\left(t\right)+\dots +c_n{P}_n^{\alpha }\left(t\right)=c_0+c_1{t}^{\alpha }+c_2\frac{1}{2}\left(3t^{2\alpha }-1\right)+\dots .$$

For example,

$$\begin{gathered} t^{2\alpha }=\frac{1}{3}{P}_0^{\alpha }\left(t\right)+0P_1^{\alpha }\left(t\right)+\frac{2}{3}{P}_2^{\alpha }\left(t\right). \\ {t}^{3\alpha }=0P_0^{\alpha }\left(t\right)+\frac{3}{5}{P}_1^{\alpha }\left(t\right)+0P_2^{\alpha }\left(t\right)+\frac{2}{5}{P}_3^{\alpha }\left(t\right). \end{gathered}$$

Example 2. 

Consider the function $f\left(t\right)=\{\begin{array}{cc}0& -1\le t<c\\ 1& c<t\le 1\end{array}$. The n-th generalized Legendre coefficient of $f$ is written as

$$\begin{array}{l}{c}_n=\frac{2n + 1}{2}\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{I}_{\alpha }^{-1}\left(f{P}_n^{\alpha }\right)\left(1\right)=\frac{2n + 1}{2}\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{I}_{\alpha }^c\left(P_n^{\alpha }\right)\left(1\right)\\ =\frac{1}{2}\frac{\mathsf{\Gamma }\left(\beta - \alpha + 1\right)}{\mathsf{\Gamma }\left(\beta \right)}{I}_{\alpha }^c\left({}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n+1}^{\alpha }\right)-{}_{ }{}^{\left(1\right)}D{ }^{GDFD }\left(P_{n-1}^{\alpha }\right)\right)\left(1\right)\\ =-\frac{1}{2}\left(P_{n+1}^{\alpha }\left(c\right)-P_{n-1}^{\alpha }\left(c\right)\right),n=1,2,3,\dots ,\end{array}$$

and $c_0=\frac{1}{2}\left(1-c^{\alpha }\right)$.

Note that in the previous developments, Equation (18), Definition 5, and Theorem 4 have been used.

Hence,

$$f\left(t\right)=\frac{1}{2}\left(1-c^{\alpha }\right)-{\displaystyle \sum }_{n=1}^{\infty }\left(P_{n+1}^{\alpha }\left(c\right)-P_{n-1}^{\alpha }\left(c\right)\right)P_n^{\alpha }\left(c\right),-1<t<1 .$$

Finally, it can be written that

$$S_m\left(c\right)=\frac{1}{2}\left(1-c^{\alpha }\right)-\frac{1}{2}{\displaystyle \sum }_{n=1}^m\left(P_{n+1}^{\alpha }\left(c\right)-P_{n-1}^{\alpha }\left(c\right)\right)P_n^{\alpha }\left(t\right)=\frac{1}{2}-\frac{1}{2}{P}_{m+1}^{\alpha }\left(c\right)P_m^{\alpha }\left(c\right),$$

and since $P_n^{\alpha }\left(c\right)\to 0$ when $n\to \infty$, it follows that

$$\underset{m\to \infty }{\mathit{lim}}{S}_m\left(c\right)=\frac{1}{2}=\frac{1}{2}\left(f\left(c^{+}\right)+f\left(c^{-}\right)\right).$$

Example 3. 

We obtain the first four generalized Legendre coefficients for $f\left(t\right)=e^{\gamma \frac{t^{\alpha }}{\alpha }}$.

$$c_0=\frac{1}{2}\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{I}_{\alpha }^{-1}\left(f{P}_0^{\alpha }\right)\left(1\right)=\frac{\alpha }{2}{\displaystyle {{\displaystyle \int }}_{-1}^1}{e}^{\gamma \frac{t^{\alpha }}{\alpha }}\frac{dt}{t^{1-\alpha }}=\frac{\alpha }{\gamma }sinh\left(\frac{\gamma }{\alpha }\right).$$

$$\begin{array}{l}{c}_1=\frac{3}{2}\frac{\alpha \mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\beta -\alpha +1\right)}{I}_{\alpha }^{-1}\left(f{P}_1^{\alpha }\right)\left(1\right)\\ =\frac{3\alpha }{2}{\displaystyle {{\displaystyle \int }}_{-1}^1}{e}^{\gamma \frac{t^{\alpha }}{\alpha }}{t}^{\alpha }\frac{dt}{t^{1-\alpha }}=3\left(\frac{\alpha }{\gamma }cosh\left(\frac{\gamma }{\alpha }\right)-{\left(\frac{\alpha }{\gamma }\right)}^{2}sinh\left(\frac{\gamma }{\alpha }\right)\right),\end{array}$$

where we had to apply integration by parts.

5. Conclusions

In this paper, we have studied the generalized fractional Legendre equation and its solutions, in the sense of the generalized Abu-Shady–Kaabar derivative. Likewise, in this context, important properties of the generalized Legendre polynomials have been established. Specifically, we have obtained classical results of Legendre polynomials such as the Rodrigues formula and some useful recurrence relations directly using this fractional derivative. We have also addressed the problem of the orthogonality of generalized Legendre polynomials, as a natural extension of the corresponding property of classical Legendre polynomials. In order to establish the orthogonality of the generalized Legendre polynomials, as in the classic case, over the interval $\left[-1, 1\right]$, it has been necessary to modify the definition of generalized $\alpha$-derivative to include negative values of the variable. Finally, for the $L_{\alpha }^2$ class functions, we have studied their representation in generalized Legendre series on the interval $\left[-1, 1\right]$. The results obtained show how this line of research represents a natural extension of the classical differential calculus. The potential of this new definition of fractional derivatives, from the theoretical point of view and its applications, is clearly shown through the developments, comments, and examples included in the previous sections. In short, this research can open the way for future work in which the results of classical mathematical analysis are expanded in the sense of this new definition of fractional derivatives.