Associated Conformable Fractional Legendre Polynomials

Along with the work of Abul-Ez et al. [37], we introduce the associated conformable fractional Legendre polynomials (ACFLPs), from which the fractional differential equation of ACFLPs is established. Subsequently, some of interesting properties are derived such as generating function, hypergeometric representation, analytical formula, besides various of recurrence relations. Also, orthogonal properties of ACFLPs are developed in conformable context. We append our study by presenting the shifted ACFLPs and driving some of important properties such as Rodrigues’ type representation formula of fractional order derivative and explicit formula. An interesting compact closed-form expression is derived from the definite integral using a convenient analytical formula for the shifted ACFLPs. This result is easily generalized for integrands involving products of an arbitrary number of shifted associated Legendre polynomials in conformable sense.


1.
Introduction Known among the Mathematicians that special functions have emerged from a wide variety of practical problems that interest not only mathematicians, it is of interest to other researchers in science to study their properties, characteristics, and applications. in general, the special functions induced as solutions of famous differential equations, such as Legendre, Laguerre, Hermite, Chebyshev, hypergeometric, etc.
Legendre's differential equation is a type of ordinary differential equation (ODE) that is commonly used in physics and engineering. It appears when equation of Laplace is solved in spherical coordinates particularly. In 1784, the significant of Legendre polynomials is sensed when the attraction of spheroids and ellipsoids was studying by A. Legendre. It is used in several areas in physics and mathematics. They may arise from solutions of Legendre ODE, such as the analog ODEs in spherical polar coordinates and the famous Helmholtz equation. Also, they appear as a result of requiring a complete, functions orthogonal sequence over the interval [-1,1]. (Gram-Schmidt orthogonalization). They are angular momentum eigen-functions in quantum mechanics.
Nowadays, the theory of fractional calculus is classified as generalized fractional integrals or derivatives in addition to its origin, as old classical calculus. Because fractional order derivatives are

Remark
We can observe the following remarks: 1) the authors in [19] found that the behaves of the CFA doing well in the rules of product and the chain unlike the case of the classic fractional calculus 2) Surprisingly enough, the CFA of the constant function is zero, however, this is not the case for the Riemann-Liouville fractional derivative. 3) when = 1 in (1), we can easily obtain the analogous ordinary classical derivatives.
Moreover, even if a function is not differentiable, it can be x-differentiable at a point; for example, if ( ) = 2√ , then 1 2 ( ) = 1. As a result, 1 2 (0) = 1. As well as, 1 (0) does not exist. This is in stark contrast to what is known about classical derivatives. 4) to solve the simple fractional differential equation , is used, then, we can observe that the general solution is = −2√ easily.
Moreover, Khalil et al. [19] introduced the -fractional integral of a real function as follows: Definition 2.1 Let : (0, ∞) → , -differentiable and ∈ (0,1], then the -fractional integral of is defined by: Neat, we'll talk about the function of Gauss hypergeometric 2 1 ( , ; ; ), by the following formula: where ( ) denotes the Pochhammer symbol, which is defined in terms of Gamma functions as: In [35], the authors introduced the conformable fractional Legendre differential equation solution as: and presented the CFLPs, ( ) as its solution. The authors in [37] gave an eaplicit formula of ( ) as follows: gave has the following eaplicit formul: 1 () Unless otherwise mentioned in the article, the fractional number is taken to have the value 0<α≤1.
In much of the work discussed here, rearranging terms in iterated sequence is a popular method. The fundamental lemma that follows is of the type that will be used to simplify the proofs in work. We have the following Lemma for the infinite double series (see [38]).
Based on the notation of conformable fractional derivative, we have = .
. . ⏟ − , as well as the fact = − , Abul-Ez et al [37], presented the following Rodrigues formula of the conformable fractional Legendre polynomials ( ) in the form: Now, we may assume that the function of associated conformable Legendre type in the form For = 0, these functions reduce to the conformable fractional Legendre polynomials defined in [37], and it is clear that if > , then ( . ) = 0.
4. Generating function, Hypergeometric representation, and the analytical formula of the associated conformable fractional Legendre polynomials In this section, we will investigate some of interesting properties of the associated conformable fractional Legendre polynomials such as generating function, hypergeometric representation, and the analytical formula. Generating function Generating functions are important way to transform formal power series into functions and to analyze asymptotic properties of sequences. The authors in [37] established the generating function of the conformable fractional Legendre polynomials as follow 1 √1−2 In what follows we characterize the associated conformable fractional Legendre polynomials ( ) by means of generating function as: Theorem 4.1 Let ∈ (0,1], ≤ , then following generating function holds true: Proof. In view of conformable derivative (1), and by differentiating relation (18), m-times, we have Replacing n by + , we have: Multiplying by (1 − 2 ) 2 , we get: Hypergeometric representation In this subsection, we introduce the associated conformable fractional Legendre polynomials (a) in terms of Gauss hypergeometric function as follow: Theorem 4.2 For ∈ (0,1], the associated Legendre polynomials (a) may be written as: Proof. The authors in [37], deduce the relation By differentiating relation (21) m-times, and using the relation:  Multiplying by (1 − 2 ) 2 , the required result is established.

Theorem 4.3
For ∈ (0,1], the following analytical formula of associated conformable fractional Legendre polynomials ( ) is true Proof. In view of (1), using the binomial expansion, we get: Using the linearity of conformable derivative, we have Therefore, as required.

5.
Recurrence Relations of Associated conformable fractional Legendre polynomials In view of (1), we establish some of recurrence relations of associated conformable fractional Legendre polynomials as follows Theorem 5.1 For ∈ (0,1], the associated conformable fractional Legendre polynomials ( ), satisfy the following recurrence relations: Along with the scalar case, we have i) This is the fundamental relationship linking three associated conformable fractional Legendre polynomials with the same and consecutive values. From equation (13), we have: Multiplying by (1 − 2 ) 2 , we get: ii) This is the fundamental relationship linking three associated conformable fractional Legendre polynomials with the same ma and consecutive values. Using conformable derivative and by differentiating () − with using Leibniz's theorem we have: Now, differentiating (7) ( − 1) − via conformable derivative, we get: Substituting from (25) in (24), we get: which is required relation. □

Orthogonality relations and expansion of functions in terms of associated CFLPs
The general theory of the expansion of analytic functions in terms of an arbitrary set of orthogonal polynomials is a basic subject in analysis, started by various authors to whom we may mention Boas [39], Faber [40], and Whittaker [41] and later on in higher dimension by Abul-Ez et al. [42,43,44,45,46]. In the case of the usual classical calculus, there are some functions that do not have Taylor power series representation about certain points but in the theory of conformable fractional calculus, they do. This fact had been shown by Abdeljawad [22], where he also stated the expansion of fractional power series for an infinity a-differentiable function within the fractional Taylor series. The expansion of a dispensed real function in a series of associated conformable fractional Legendre polynomials is useful in various applications, for example, in giving the solution of certain fractional differential equations.
In this section, we introduced the orthogonality relations of the associated conformable fractional Legendre polynomials and then use it to establish the expansion of a real valued functions in terms of associated CFLPs.

6.1.
Orthogonality It is well known that the orthogonality relations play an important rule in many applications. The authors in [37] deduced the orthogonality relations of conformable fractional Legendre polynomials as follows: where ′ is a famiular kronker delta and = −1 .

7.1.
Rodrigues formula for shifted ACFLPs Rodrigues' formula is considered a most of important method to define a orthogonal polynomials sequence [48], and it can be derived a good properties if it is known. For this reason, the mathematicians have been focused on these formulas' generalizations in last two decades, both new classes of special functions and polynomials are defined, as well as fractional order differentiation is included. Rajković and Kiryakova [47] discussed and defined the special functions based on a Rodrigues formula for shifted Legendre polynomials. Moreover, they tested the orthogonality property, which only applied in certain situations. The Rodrigues formula for Shifted Conformable Fractional Legendre Polynomials (SCFLP) was also introduced by authors in [37] as: Now, we will establish the Rodrigues formula for shifted associated conformable fractional Legendre polynomials.
The following theorem, in addition to formula (10), gives the Rodrigues formula for shifted ACFLPs.

Theorem 7.1
The shifted associated conformable fractional Legendre polynomials ̃( ) can be written in the sense of conformable derivative as where ̃( ) is the (SCFLP).
Proof. Directly, by taking the transformation = 2 − 1 in (10), we can establish the required result.

7.3.
Overlap integral of shifted ACFLPs There has been interest in driving closed expression for the definite integral ( 1 , 1 ; 2 , 2 ) = ∫ 1 0̃1 involving a product of two shifted ACFLPs.
In this subsection, one can use the analytical form of ̃( ) to drive a simple closed expression of the integral (36 Integral (38) can be easily evaluated by noting that it possesses the form of the beta function, we have: Equation (41) is thus the analytical solution of the integral (36) which involves summing terms of alternating sign. For example, we can evaluate the integral (36) with various (arbitrary) values of 1 , 1 , 2 , 2 and as shown in table 1. It is clear that this method can be easily generalized in a straightforward manner for integrals containing a product of any finite number of associated conformable fractional Legendre polynomials.

8.
Conclusion In view of many practical problems that interest not only for mathematicians and other researchers in sciences, it was found that special functions induced as solutions of famous differential equations, like as Legendre, Bessel, Laguerre, Hermite, hypergeometric, etc. Special functions of fractional calculus emerged and drew a lot of attention because of their wide range of applications in science and engineering. It is known that Legendre polynomials play an important role in solving many differential equations which are related to real life phenomena.
Motivated by the applications involved Legendre differential equation, the current paper introduced the associated Legendre polynomials in the conformable fractional sense. Some interesting properties such as hypergeometric representation, analytical formula, various recurrence relations are established. Orthogonal properties of such associated conformable fractional Legendre polynomials are developed. An interesting compact closed-form expression is derived from the definite integral using a convenient analytical formula for the shifted ACFLPs. All obtained results are newly presented and can be extended as well as they are useful for applications.