Left- and Right-Shifted Fractional Legendre Functions with an Application for Fractional Differential Equations

Two new orthogonal functions named the leftand the right-shifted fractional-order Legendre polynomials (SFLPs) are proposed. Several useful formulas for the SFLPs are directly generalized from the classic Legendre polynomials. The left and right fractional differential expressions in Caputo sense of the SFLPs are derived. As an application, it is effective for solving the fractional-order differential equations with the initial value problem by using the SFLP tau method.


Introduction
Legendre polynomials are a family of complete and orthogonal functions discovered by Adrien-Marie Legendre in 1782. As a very important application, Legendre spectral methods are successfully used to obtain numerical solutions of the various differential equations. Through Google Scholar search, there are almost 54,000 articles from 1980 to 2019 on the use of the Legendre spectral methods in the study of various problems, such as numerical solving for integrodifferential equations ( [1][2][3][4][5][6]) and ordinary differential equations with fractional order ( [7][8][9]) and integer order ( [10]). Recently, the Legendre spectral method was proved to be an effective method to solve fractional differential equations, which has been studied by many scholars ( [7,8,[11][12][13]). More recently, many authors ( [14][15][16][17][18][19]) applied Müntz orthogonal polynomials to solve the fractionalorder differential equations (FDEs). Motivated by this literature, we define the left SFLPs by introducing the change of variable z L = 2ððx − aÞ/ðb − aÞÞ α − 1. In particular, when a = −1, b = 1, and α = 1, the left SFLPs degenerate the classic Legendre polynomials; while a = 0, b = 1, and 0 < α < 1, the left SFLPs are transformed into the fractional-order Legendre polynomials proposed in [7]. Similarly, the right SFLPs can be also obtained by introducing the change of variable z R = −2ððb − xÞ/ðb − aÞÞ α + 1. Furthermore, to solve some FDEs, the SFLP tau method is better than the method based on the other orthogonal polynomials.

Shifted Fractional-Order Legendre Polynomials
In this section, we introduce some definitions, notations, and useful formulas about the shifted fractional-order Legendre polynomials. For the properties of classical Legendre polynomials, please refer to the literature [7,11]. Now, we begin with the definition of Caputo fractional derivative.
Definition 1. (see [20]). For m − 1 < α ≤ m, m ∈ Z + , a, b ∈ R, the left side and the right side Caputo fractional derivative is defined by Then, for α, β > 0 and constant C, we have the following properties: Next, let us recall the definition of the classic Legendre polynomials. The classic Legendre polynomials, denoted by L n ðzÞ, n = 0,1,…, are orthogonal on the interval ½−1, 1 with the orthogonality property where δ nm is the Kronecker function. Now, in order to apply Legendre polynomials on the finite interval ½a, b, we define the left and right SFLPs by introducing the change of variable z = z L and z = z R , respectively. Then, these two functions, denoted by LL α n ðxÞ and RL α n ðxÞ, n = 0,1,…, are orthogonal polynomials with the weight function w L ðxÞ = ðx − aÞ α − 1 and w R ðxÞ = ðb − xÞ α − 1 , respectively, those are Let a = −1, b = 1, we plot the first six terms of the left and right SFLPs for α = 0:5 in Figure 1, and in Figure 2 (1) The analytic forms of the left SFLPs and the right SFLPs (2) Three-term recurrence relations for the left SFLPs with LL α 0 ðxÞ = 1 and LL α 1 ðxÞ = 2ððx − aÞ/ðb − aÞÞ α − 1, and the right SFLPs with RL α 0 ðxÞ = 1 and RL α 1 ðxÞ = −2ððb − xÞ/ðb − aÞÞ α + 1 (3) Derivative recurrence relations for the left SFLPs and the right SFLPs (4) The boundary values of the left and right SFLPs (5) The left and right Legendre's differential equations of fractional order 2

Advances in Mathematical Physics
In the following lemmas, we derive the fractional differential expressions of the left and right SFLPs in Caputo sense.

Lemma 2. Let α > 0 and
Then, we have Lc n,i Lc m,j where Lc n,i and Lc m,j are given by (7).

2), and (3) lead to
From Lemma 2, it is elementary to get with Ld α i,j given by (17). The following lemma on fractional differential expressions for the right SFLPs can be obtained similarly.

Lemma 3. Let α > 0 and
Then, we have where Rc n,i and Rc m,j are given by (8).

Application
In this section, we give two examples to illustrate that our methods are effective. First, we apply the left SFLP tau method to solve the fractional-order differential equation of the following form: Suppose f ðxÞ = ðΓð2α + 1Þ/Γðα + 1ÞÞ ⋅ ðx + 1Þ α + ðx + 1Þ 2α . Then, the exact solution of (21) is uðxÞ = ðx + 1Þ 2α . Now, we use the left SFLP tau method to obtain it. Let with C T = ½c 0 , c 1 , ⋯, c n − 1 and ϕðxÞ T = ½LL α 0 ðxÞ, LL α 1 ðxÞ, ⋯, LL α n−1 ðxÞ. From Lemma 2, we have with the matrix D α n = fLd i,j g n×n , where Ld i,j is given by (17). Assume with Set n = 3. Then By (22), (23), and (24), we have with three-order matrix Advances in Mathematical Physics which, in accordance with uð−1Þ = ∑ 2 i=0 c i LL α i ð−1Þ = c 0 − c 1 + c 2 = 0, yields Using (22), we obtain the exact solution of (21). Obviously, we cannot get this exact solution by the classic Legendre tau method.

Conclusions
In this paper, the left-and right-shifted fractional-order Legendre polynomials are proposed by substituting the variables of the classic Legendre polynomials. Correspondingly, the differential expressions of these new polynomials for the left and right fractional derivatives in Caputo sense are derived, based on which the tau method can be used to solve the FDEs on the arbitrary finite interval ½a, b. Moreover, the results in this article are easy to generalize to the case of the other orthogonal functions, e.g., Chebyshev polynomials, which will be studied later.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.